Chapter 4

Applications of Differential Calculus to Analysis

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Section 4.1The Theorems of Rolle, Lagrange, and Cauchy. Increase and Decrease of Functions. Inequalities[36/6%]

Problem 1385

Prove that the roots of the derivative of the polynomial $y = x(x-1)(x-2)(x-3)(x-4)$ are real, and indicate the bounds between which they lie.

Problem 1386

Prove that the derivative of a polynomial all of whose roots are real has no imaginary roots.

Problem 1387

Prove that the Legendre polynomial

P_{n}(x) = \frac{1}{2^{n} \cdot n!} \frac{d^{n}(x^{2}-1)^{n}}{dx^{n}}

has all its roots real and lying in the interval $(-1, 1)$ .

Problem 1388

Prove that the Laguerre polynomial

L_{n}(x) = e^{x} \frac{d^{n} x^{n} e^{-x}}{dx^{n}}

has all its roots positive.

Problem 1389

Prove that the Hermite--Chebyshev polynomial $e^{x^{2}} \dfrac {d^{n} e^{-x^{2}}}{dx^{n}}$ has all its roots real.

Problem 1390

There is an equality $\dfrac {d^{n} \arctg x}{dx^{n}} = \dfrac {P_{n-1}(x)}{(1+x^{2})^{n}}$ , where $P_{n-1}(x)$ is a polynomial of degree $n-1$ . Prove that all the roots of this polynomial are real.

Problem 1391

For what values of $x$ is the function $\dfrac {x}{1+x^{2}}$ increasing?

Problem 1392

For what values of $x$ is the function $x^{3}(1-x)$ decreasing?

Problem 1393

Prove that the function $x^{n}e^{-x}$ , where $n > 0$ , increases for $0 < x < n$ and decreases for $x > n$ .

Problem 1394

Prove that for $0 < x < \pi$ the function $\dfrac {\sin x}{x}$ is decreasing.

Problem 1395

Prove that as the number of sides increases, the perimeter of a regular inscribed polygon increases, while the perimeter of a circumscribed one decreases.

Problem 1396

Using theorems on the increase of functions, prove that $\left(1+\dfrac {1}{n}\right)^{n}$ increases for all $n > 0$ .

Problem 1397

Using Lagrange's formula, prove the inequality $n(b-a)a^{n-1} < b^{n}-a^{n} < n(b-a)b^{n-1}$ for $b > a > 0$ , $n > 1$ .

Problem 1398

Using Lagrange's formula, prove the inequality $\dfrac {x}{1+x} < \ln (1+x) < x$ for $x > 0$ .

Problem 1399

Using Lagrange's formula, prove the inequality $e^{x} > 1+x$ .

Problem 1400

Using Lagrange's formula, prove the inequality $e^{x} > ex$ for $x > 1$ .

Problem 1401

Using Lagrange's formula, prove the inequality $x^{\delta }\left|\ln x\right| < \dfrac {1}{\delta e}$ for $0 < x < 1$ , $\delta > 0$ .

Problem 1402

Prove the theorem: if at $x=0$ the functions $\varphi (x), \varphi '(x), \ldots , \varphi^{(n-1)}(x)$ vanish, and the function $\varphi^{(n)}(n)$ is positive for $x > 0$ , then $\varphi (x) > 0$ for positive $x$ .

Problem 1403

Prove the theorem: if

\varphi (0) = \psi (0), \; \varphi '(0) = \psi '(0), \; \ldots , \; \varphi ^{(n-1)}(0) = \psi ^{(n-1)}(0)

and $\varphi^{(n)}(x) > \psi^{(n)}(x)$ for $x > 0$ , then for positive $x$ the inequality $\varphi (x) > \psi (x)$ holds.

Problem 1404

Prove the inequality $x-\dfrac {x^{3}}{6} < \sin x < x$ , for $x > 0$ .

Problem 1405

Prove the inequality $x-\dfrac {x^{3}}{3} < \arctg x < x$ , for $x > 0$ .

Problem 1406

Prove the inequality $\tg x > x+\dfrac {x^{3}}{3}$ , for $0 < x < \dfrac {\pi }{2}$ .

Problem 1407

Prove the inequality $x-\dfrac {x^{2}}{2} < \ln (1+x) < x$ ; $x > 0$ .

Problem 1408

By direct squaring, prove the inequalities

1+\frac{x}{2}-\frac{x^{2}}{8} < \sqrt{1+x} < 1+\frac{x}{2} \quad \text{for } 0 < x < 8.

Problem 1409

Prove that for $0 < x < 1$ the quantity $e^{2x}$ is less than $\dfrac {1+x}{1-x}$ .

Problem 1410

Prove that if $p_{n}$ is the perimeter of a regular inscribed polygon, and $P_{n}$ that of a regular circumscribed polygon, then $\dfrac {2}{3}p_{n}+\dfrac {1}{3}P_{n}$ is greater than the length of the circle.

Remark.

Author's note: the result is due to Huygens.

Problem 1411

Prove that for a geometric progression with positive terms, the sum of terms equidistant from the ends of the progression is less than the sum of the extreme terms.

Problem 1412

An arithmetic progression and a geometric progression have the same number of terms and the same extreme terms respectively, and all terms of both progressions are positive. Show that the sum of the terms of the arithmetic progression is greater than that of the geometric progression.

Problem 1413

The numbers $m_{1}, m_{2}, \ldots , m_{n}$ are positive, and the points

(x_{1}, y_{1}), \quad (x_{2}, y_{2}), \quad \ldots , \quad (x_{n}, y_{n})

lie on the curve $y = \varphi (x)$ . Prove that if the curve is convex downward, then the inequality

\varphi \left(\frac{m_{1}x_{1}+m_{2}x_{2}+\ldots +m_{n}x_{n}}{m_{1}+m_{2}+\ldots +m_{n}}\right) < \frac{m_{1}\varphi (x_{1})+m_{2}\varphi (x_{2})+\ldots +m_{n}\varphi (x_{n})}{m_{1}+m_{2}+\ldots +m_{n}}.

holds. If, on the other hand, the curve is convex upward, then the sign of the inequality is reversed.

Problem 1414

The following problems (1414--1418) reduce to problem 1413. The numbers $\alpha_{1}, \alpha_{2}, \ldots , \alpha_{n}$ occurring in them are positive. The numbers $x_{1}, x_{2}, \ldots , x_{n}$ are assumed positive and such that not all of them are equal.

Prove that for $m > 1$ the inequality

(\alpha _{1}x_{1}+\alpha _{2}x_{2}+\ldots +\alpha _{n}x_{n})^{m} < \alpha _{1}x_{1}^{m}+\alpha _{2}x_{2}^{m}+\ldots +\alpha _{n}x_{n}^{m},

holds, where $\alpha_{1}+\alpha_{2}+\ldots +\alpha_{n}=1$ .

Problem 1415

Prove that the arithmetic mean of positive numbers is less than the quadratic mean of the same numbers, i.e., that

\frac{x_{1}+x_{2}+\ldots +x_{n}}{n} < \sqrt{x_{1}^{2}+x_{2}^{2}+\ldots +x_{n}^{2}}.

Problem 1416

Prove that for $m > 1$ and positive numbers $x_{1}, x_{2}, \ldots , x_{n}$ the inequality

(x_{1}+x_{2}+\ldots +x_{n})^{m} < n^{m-1}(x_{1}^{m}+x_{2}^{m}+\ldots +x_{n}^{m}).

holds.

Problem 1417

If the numbers $\alpha$ and the numbers $x$ are positive and $\alpha_{1}+\alpha_{2}+\ldots +\alpha_{n}=1$ , then the inequality

\alpha _{1}x_{1}+\alpha _{2}x_{2}+\ldots +\alpha _{n}x_{n} > x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}} \ldots x_{n}^{\alpha _{n}}.

holds. In particular, the arithmetic mean of positive numbers is greater than the geometric mean of the same numbers. Prove this.

Problem 1418

For any real values of the coefficients $a_{\nu }$ and $b_{\nu }$ , as well as of the variable $x$ , the obvious inequality

\sum _{\nu =1}^{n}(a_{\nu }x+b_{\nu })^{2} \geq 0.

holds. Starting from it, derive the Cauchy inequality,

(a_{1}b_{1}+a_{2}b_{2}+\ldots +a_{n}b_{n})^{2} \leq (a_{1}^{2}+a_{2}^{2}+\ldots +a_{n}^{2})(b_{1}^{2}+b_{2}^{2}+\ldots +b_{n}^{2}).

Problem 1419

In the following problems (1419--1420) it is useful to apply a change of coordinates.

Prove that on the circle $x^{2}+y^{2}=1$ the inequality

\frac{1}{a^{2}} \leq Ax^{2}+Bxy+Cy^{2} \leq \frac{1}{b^{2}},

holds, if the curve $Ax^{2}+Bxy+Cy^{2}=1$ is an ellipse with semi-axes $a$ and $b$ .

Problem 1420

The surface $Ax^{2}+By^{2}+Cz^{2}+Dxy+Exz+Fyz=1$ , after a suitable rotation of axes, has the equation $A_{1}x_{1}^{2}+B_{1}y_{1}^{2}+C_{1}z_{1}^{2}=1$ , where $A_{1} < B_{1} < C_{1}$ . Prove the inequality

A_{1} \leq \frac{Ax^{2}+By^{2}+Cz^{2}+Dxy+Exz+Fyz}{x^{2}+y^{2}+z^{2}} \leq C_{1}.

Section 4.2Finding the Greatest and Least Values of Functions of One Variable[43/95%]

Problem 1421

Find the maximum of the function $y = 6x-x^{2}$ .

Problem 1422

Find the minimum of the function $y = x^{2}-8x$ .

Problem 1423

Find the extreme values of the function $y = x^{3}-12x$ .

Problem 1424

Show that the function $y = x^{3}(8-x)$ has a maximum at $x=6$ and has no extremum at $x=0$ .

Problem 1425

Show that the function $y = x^{2}+\dfrac {1}{2}x^{2}\sin \dfrac {1}{x}$ has a minimum at $x=0$ , although the first derivative, as $x$ passes through zero, changes sign neither from $-$ to $+$ nor from $+$ to $-$ .

Problem 1426

Find the extreme values of the function $y = x^{3}-3x^{2}+6x+7$ .

Problem 1427

Find the extreme values of the function $y = x^{3}-9x^{2}+15x-3$ .

Problem 1428

Find the extreme values of the function $y = a+(x-b)^{4}$ .

Problem 1429

Find the extreme values of the function $y = a+(x-b)^{3}$ .

Problem 1430

Find the extreme values of the function $y = x^{4}-8x^{3}+22x^{2}-24x+12$ .

Problem 1431

Find the extreme values of the function $y = x^{5}-5x^{4}+5x^{3}-1$ .

Problem 1432

Find the extreme values of the function $y = (x-4)^{4}(x+3)^{3}$ .

Problem 1433

Find the extreme values of the function $y = \dfrac {x}{1+x^{2}}$ .

Problem 1434

Find the extreme values of the function $y = x+\dfrac {1}{x}$ .

Problem 1435

Find the extreme values of the function $y = \dfrac {a^{2}}{x}+\dfrac {b^{2}}{a-x}$ .

Problem 1436

Find the extreme values of the function $y = \dfrac {x^{2}-7x+6}{x-10}$ .

Problem 1437

Find the extreme values of the function $y = \dfrac {x^{4}+1}{x^{2}}$ .

Problem 1438

Find the extreme values of the function $y = \dfrac {x^{2}}{x^{4}+4}$ .

Problem 1439

Find the extreme values of the function $y = x\ln x$ .

Problem 1440

Find the extreme values of the function $y = x^{2}\ln x$ .

Problem 1441

Find the extreme values of the function $y = x^{x}$ .

Problem 1442

Find the extreme values of the function $y = x\ln^{2} x$ .

Problem 1443

Find the extreme values of the function $y = x^{n}e^{-x}$ , $n$ positive.

Problem 1444

Find the extreme values of the function $y = x^{2}e^{-x^{2}}$ .

Problem 1445

Find the extreme values of the function $y = e^{x}+e^{-x}$ .

Problem 1446

Find the extreme values of the function $y = e^{-x}-e^{-2x}$ .

Problem 1447

Find the extreme values of the function $y = x^{3}\sqrt[3]{(x-1)^{2}}$ for $-2 \leq x \leq 2$ .

Problem 1448

Find the extreme values of the function $y = \ln x-\arctg x$ .

Problem 1449

Find the extreme values of the function $y = e^{x}\cos x$ .

Problem 1450

Find the extreme values of the function $y = \dfrac {\sin x}{1-a^{2}\cos^{2}x}$ for $a^{2} < 1$ (consider two cases: $2a^{2} > 1$ and $2a^{2} < 1$ ).

Problem 1451

Find the extreme values of the function $y = \dfrac {\tg x}{\tg (x+a)}$ for $0 < a < \dfrac {\pi }{2}$ .

Problem 1452

Find the extreme values of the function $y = \dfrac {\tg^{3} x}{\tg 3x}$ .

Problem 1453

Find the extreme values of the function $y = \sin 3x-3\sin x$ .

Problem 1454

Find the extreme values of the function $y = \arcsin \sin x$ .

Problem 1455

Find the extreme values of the function $y$ given implicitly by the equation $y^{2}+2yx^{2}+4x-3=0$ .

Problem 1456

Find the extreme values of the function $y$ given implicitly by the equation $y^{2}+2yx^{2}-4x-3=0$ .

Problem 1457

Find the extreme values of the function $y$ given implicitly by the equation $xy^{2}-x^{2}y=2a^{3}$ .

Problem 1458

Find the extreme values of the function $y$ given implicitly by the equation $x^{2}+2xy+y^{2}-4x+2y-2=0$ .

Problem 1459

Find the extreme values of the function $y$ given implicitly by the equation $x^{2}-2xy+5y^{2}-2x+4y+1=0$ .

Problem 1460

Find the extreme values of the function $y$ given implicitly by the equation $x^{2}+4xy+4y^{2}+x+2y-1=0$ .

Problem 1461

Find the extreme values of the function $y$ given implicitly by the equation $x^{3}+y^{3}-3axy=0$ for $a > 0$ .

Problem 1462

Find the extreme values of the function $y$ given implicitly by the equation $x^{4}+y^{4}-4xy=0$ .

Problem 1463

Find the extreme values of the function $y$ given implicitly by the equation $x^{4}+y^{4}=x^{2}+y^{2}$ .

Section 4.3Plotting Graphs of Functions[57/47%]

Problem 1464

Plot the graph of the function $y = x^{3}-3a^{2}x$ .

Problem 1465

Plot the graph of the function $y = \dfrac {ax^{2}}{a^{2}+x^{2}}$ .

Problem 1466

Plot the graph of the function $y = (x+1)^{2}(x-2)$ .

Problem 1467

Plot the graph of the function $y = \dfrac {x-1}{x^{2}-3x-4}$ .

Problem 1468

Plot the graph of the function $y = \dfrac {a^{3}}{a^{2}+x^{2}}$ .

Problem 1469

Plot the graph of the function $y = Ae^{-x^{2}/a^{2}}$ , $A > 0$ .

Problem 1470

Plot the graph of the function $y = x^{n}e^{-x/a}$ , $n > 0$ , $a > 0$ .

Problem 1471

Plot the graph of the function $y = x^{2}e^{-x^{2}/a^{2}}$ .

Problem 1472

Plot the graph of the function $y = \dfrac {1}{x^{2}}-\dfrac {1}{(x-1)^{2}}$ .

Problem 1473

Plot the graph of the function $y = \dfrac {x^{2}}{2}+\dfrac {1}{x}$ .

Problem 1474

Plot the graph of the function $y = \dfrac {1}{x}+\dfrac {2x}{x^{2}-1}$ .

Problem 1475

Plot the graph of the function $y = \dfrac {2x-1}{(x-1)^{2}}$ .

Problem 1476

Plot the graph of the function $y = \dfrac {x^{2}+1}{x^{2}-4x+3}$ .

Problem 1477

Plot the graph of the function $y = \pm \sqrt{\dfrac {x-1}{x+1}}$ .

Problem 1478

Plot the graph of the function $y = \dfrac {10\sqrt[3]{(x-1)^{2}}}{x^{2}+9}$ .

Problem 1479

Plot the graph of the function given by the equation $2y = \sqrt{x^{2}+x+1}+\sqrt{x^{2}-x+1}$ .

Problem 1480

Plot the graph of the function $y = \sqrt{x^{2}+x+1}-\sqrt{x^{2}-x+1}$ .

Problem 1481

Plot the graph of the function given by the equation $y = \pm x^{2}\sqrt{x+1}$ .

Problem 1482

Plot the graph of the function $y = (x+1)^{3}\sqrt[3]{x^{2}}$ .

Problem 1483

Plot the graph of the function $y = \sqrt[3]{(x+1)^{2}}+\sqrt[3]{(x-1)^{2}}$ .

Problem 1484

Plot the graph of the function $y = \sqrt[3]{(x+1)^{2}}-\sqrt[3]{(x-1)^{2}}$ .

Problem 1485

Plot the graph of the function $y = \sqrt{x^{2}+1}-\sqrt{x^{2}-1}$ .

Problem 1486

Plot the graph of the function given by the equation $y = \pm \dfrac {x}{\sqrt{x+1}}$ .

Problem 1487

Plot the graph of the function given by the equation $y = \pm \sqrt{\dfrac {1-x}{x}}$ .

Problem 1488

Plot the graph of the function given by the equation $y = \pm \dfrac {x}{\sqrt{1-x^{2}}}$ .

Problem 1489

Plot the graph of the function given by the equation $y = \pm \dfrac {x\sqrt{1-x}}{1+x}$ .

Problem 1490

Plot the graph of the function given by the equation $ay = \pm x\sqrt{x(a-x)}$ , $a > 0$ .

Problem 1491

Plot the graph of the function given by the equation $y^{2} = x^{3}+px+q$ for $4p^{3}+27q^{2} > 0$ .

Problem 1492

Plot the graph of the function given by the equation $y^{2} = x^{3}+px+q$ for $4p^{3}+27q^{2} < 0$ .

Problem 1493

Plot the graph of the function given by the equation $y^{2} = x^{3}+px+q$ for $4p^{3}+27q^{2} = 0$ .

Problem 1494

Plot the graph of the function $y = \cos^{3}x+\sin^{3}x$ .

Problem 1495

Plot the graph of the function $y = \cos^{4}x+\sin^{4}x$ .

Problem 1496

Plot the graph of the function $y = \cos x+\dfrac {1}{2}\cos 2x+\dfrac {1}{3}\cos 3x$ .

Problem 1497

Plot the graph of the function $y = \sin x+\dfrac {1}{2}\sin 2x+\dfrac {1}{3}\sin 3x$ .

Problem 1498

Plot the graph of the function $y = \dfrac {\cos 2x}{\cos x}$ .

Problem 1499

Plot the graph of the function $y = \dfrac {\tg 3x}{\tg x}$ .

Problem 1500

Plot the graph of the function $y = \dfrac {\sin x}{x}$ .

Problem 1501

Plot the graph of the function $y = \sin x^{2}$ .

Problem 1502

Plot the graph of the function $y = x\ln x$ .

Problem 1503

Plot the graph of the function $y = x^{2}\ln x$ .

Problem 1504

Plot the graph of the function $y = \ln (x^{2}-1)$ .

Problem 1505

Plot the graph of the function $y = \dfrac {1}{2}\ln \dfrac {1+x}{1-x}$ .

Problem 1506

Plot the graph of the function $y = (1+x)^{1/x}$ for $x \geq -1$ .

Problem 1507

Determine the number of real roots of the equation $12x^{4}-14x^{3}-3x^{2}-5=0$ .

Problem 1508

Determine the number of real roots of the equation $x^{4}-4ax^{3}-2=0$ .

Problem 1509

Determine the number of real roots of the equation $2x^{3}-3ax^{2}+1=0$ .

Problem 1510

Determine the number of real roots of the equation $x\ln x=a$ .

Problem 1511

Determine the number of real roots of the equation $\ln x=ax$ .

Problem 1512

Prove that the equation $x^{3}+px+q=0$ with real $p$ and $q$ has one real root when $4p^{3}+27q^{2} > 0$ and three real roots when $4p^{3}+27q^{2} < 0$ .

Problem 1513

Prove that for $a > 1$ the equation $a^{x}=bx$ has two real roots when $b > e\ln a$ , has no real roots when $e\ln a > b > 0$ , and has one root when $b < 0$ .

Problem 1514

Determine for which value of the parameter $m$ the equation $3x^{4}+4x^{3}-6x^{2}-12x+m=0$ has two distinct roots.

Problem 1515

Determine for which value of the parameter $m$ the equation $2x^{3}-13x^{2}-20x+m=0$ has one root.

Problem 1516

Determine for which value of the parameter $m$ the equation $3x^{4}-14x^{3}-45x^{2}+m=0$ has four distinct roots.

Problem 1517

Determine for which value of the parameter $m$ the equation $2x^{3}-4x^{2}-30x+m=0$ has two coincident roots and one simple root.

Problem 1518

Determine for which value of the parameter $m$ the equation $x^{2}+x+e^{-x}+m=0$ has two coincident roots.

Problem 1519

Determine for which value of the parameter $m$ the equation $x^{2}-x-\ln x+m=0$ has no roots at all.

Problem 1520

Determine for which value of the parameter $m$ the equation $6\arctg x-x^{3}+m=0$ has three roots, two of which coincide.

Section 4.4Miscellaneous Problems on Greatest and Least Values[64/98%]

Problem 1521

Determine the greatest area of a rectangle with perimeter $4a$ .

Problem 1522

Determine the greatest area of a rectangle inscribed in a circle of radius $a$ .

Problem 1523

Determine the greatest area of a rectangle inscribed in a segment of a circle of radius $a$ , if the central angle subtended by the segment equals $2\alpha$ .

Problem 1524

Determine the greatest area of a rectangle inscribed in a parabolic segment bounded by the parabola $y^{2}=2px$ and the line $x=\dfrac {m^{2}}{2p}$ .

Problem 1525

Determine the greatest area of a rectangle one side of which lies on the base $a$ of a given triangle, while two vertices lie on the lateral sides of the triangle, if the triangle has height $h$ .

Problem 1526

Four equal squares are cut from the corners of a rectangular sheet with sides $a$ and $b$ . A box is formed from the remaining cross-shaped figure, whose height equals the side of the square. Find the side length of the cut-out square for which the volume of the box is greatest.

Problem 1527

The cross-section of a log is a circle of radius $a$ . A beam with a rectangular cross-section is hewn from the log. The strength of the beam is proportional to the base and the square of the height of the cross-section. Find the shape of the cross-section for which the beam has the greatest strength.

Problem 1528

Find the greatest area of a rectangle inscribed symmetrically in a sector of a circle of radius $a$ with central angle $2\alpha$ .

Problem 1529

Find the greatest volume of a cylinder inscribed in a sphere of radius $a$ .

Problem 1530

Find the greatest volume of a cylinder for which the perimeter of the axial cross-section equals $6m$ .

Problem 1531

Find the greatest lateral surface of a cylinder inscribed in a sphere of radius $a$ .

Problem 1532

Find the greatest volume of a cylinder inscribed in a given cone.

Problem 1533

Find the greatest volume of a cylinder inscribed in a segment of the paraboloid $az=x^{2}+y^{2}$ , $a > 0$ , bounded by the plane $z=h>0$ .

Problem 1534

Find the greatest volume of a cone inscribed in a sphere of radius $a$ .

Problem 1535

Find the least volume of a cone circumscribed about a hemisphere of radius $a$ .

Problem 1536

Find the least volume of a cone circumscribed about a sphere of radius $a$ .

Problem 1537

Find the greatest volume of a cylinder whose axis runs along the diagonal of a cube with edge $a$ , and whose bases are tangent to the faces of the cube.

Problem 1538

Find the greatest volume of a cylinder whose axis intersects at a right angle the axis of a given cylinder of radius $a$ , and whose bases are tangent to the lateral surface of the given cylinder.

Problem 1539

Find the greatest volume of a cone with given slant height $l$ .

Problem 1540

A conical funnel is rolled up from a sector of a circle of given radius. For what central angle does it have the greatest volume?

Problem 1541

Find the least lateral surface of a cone having given volume $V$ .

Problem 1542

The perimeter of an isosceles triangle equals $2p$ . What must its sides be so that the volume of the solid obtained by rotating this triangle about its base is greatest.

Problem 1543

A cylinder is capped on top with a hemisphere of the same radius. The volume of the whole solid, $V$ , is given. For what radius will the total surface of the solid be least?

Problem 1544

Points move uniformly along the coordinate axes with speeds $v_{1}$ and $v_{2}$ . At the initial moment they occupied the positions $(a, 0)$ and $(0, b)$ . Find the shortest distance between the points.

Problem 1545

Find the shortest distance from the point $(x_{1}, y_{1}, z_{1})$ to the points of the line

x = a+lt, \quad y = b+mt, \quad z = c+nt.

Problem 1546

A point travels in medium I with speed $v_{1}$ , and in medium II with speed $v_{2}$ . The interface between the media is a straight line. Prove that the motion of the point from a point $A$ in one medium to a point $B$ in the other medium takes place in the shortest time provided the path consists of the straight segments $AC$ and $CB$ , where $C$ lies on the boundary of the two media, with $\dfrac {\cos \varphi_{1}}{\cos \varphi_{2}} = \dfrac {v_{1}}{v_{2}}$ , where $\varphi_{1}$ and $\varphi_{2}$ are the angles formed by the lines $AC$ and $CB$ with the interface.

Problem 1547

A point moves in the plane with speed $v_{1}$ , and upon reaching the axis $Ox$ can move with speed $v_{2} > v_{2}$ . Find the fastest path from the point $A(0, a)$ to the point $B(b, 0)$ .

Problem 1548

A canal of width $b$ branches off at a right angle from a canal of width $a$ . The walls of the canals are straight right up to the vertex of the angle. Find the greatest length $l$ of a log that can be floated along these canals from one into the other.

Problem 1549

A lever of the second kind has its fulcrum at one end and is balanced by a force $f$ at the other end. A load $p$ is hung at distance $a$ from the fulcrum, and the weight per unit length of the lever equals $m$ . Determine the length of the lever $x$ so that the force $f$ is least.

Problem 1550

From points $A$ and $A_{1}$ , two bodies set out simultaneously along the straight lines $AO$ and $A_{1}O$ toward the point $O$ , with speeds $v$ and $v_{1}$ . Here $AO = l$ , $A_{1}O = l_{1}$ , and the angle between $AO$ and $A_{1}O$ equals $\alpha$ . When is the distance between the bodies least?

Problem 1551

A cup has the shape of a hemisphere of radius $a$ . A rod of length $l > 2a$ is lowered into it. In what position of the rod is its midpoint lowest (equilibrium position)?

Problem 1552

A rod of length $2b$ rests with its ends on two lines in a vertical plane, inclined to the horizontal at angles $\alpha$ and $\beta$ . In what position of the rod is its midpoint highest?

Problem 1553

A luminous point lies on the line joining two spheres. For what position of it will the sum of the illuminated parts of the surfaces of the spheres be greatest?

Problem 1554

The brightness of illumination is expressed by the formula $f = \dfrac {m\sin \varphi }{r^{2}}$ , where $\varphi$ is the angle of inclination of the rays, $r$ is the distance from the surface element to the light source, and $m$ is a constant (the intensity of the light source). At what height $h$ should a lamp be placed on a post so that the illumination of a horizontal surface at distance $a$ from the post is greatest?

Problem 1555

A vessel with a vertical wall of height $h$ stands on a horizontal plane. A jet issues from an opening in the wall. Determine the position of the opening for which the range of the jet is greatest, if the speed of the outflowing liquid, by Torricelli's law, equals $\sqrt{2gx}$ , where $x$ is the depth of the opening.

Problem 1556

At what inclination of the lateral sides of an isosceles trapezoid is its area greatest, if the lateral sides equal $b$ and the shorter base equals $a$ .

Problem 1557

A semicircle is constructed on one side of a rectangle. The perimeter of the whole resulting figure is given. Find the greatest possible area of this figure.

Problem 1558

On a page of a book, the printed text must occupy area $S$ ; the top and bottom margins must be of width $a$ , the right and left ones of width $b$ . For what ratio of width to height of the text will the area of the whole page be least?

Problem 1559

Find the internal dimensions of an open cylindrical vessel of given capacity $V$ with wall thickness $a$ , requiring the minimum amount of material.

Problem 1560

Determine the greatest possible difference between the angles that the lateral edge and the apothem of a regular $n$ -gonal pyramid make with the plane of the base.

Problem 1561

From what point of the axis $Ox$ is the segment on the axis $Oy$ lying between the points $(0, h)$ and $(0, H)$ seen at the greatest angle ( $H > h > 0$ )?

Problem 1562

Through a point $A$ inside an angle, draw a line $MN$ so that the sum of the segments of the sides $OM+ON$ is least.

Problem 1563

Through a point inside an angle, draw a line that cuts off from the angle a triangle of least area.

Problem 1564

The upper base of a regular hexagonal prism is the hexagon $ABCDEFA$ . Through a point $O$ on the axis of the prism, lying above the upper base, and through the diagonals $AC$ , $CE$ and $EA$ , three planes are drawn, intersecting the lateral edges of the prism not passing through $A, C$ and $E$ at the points $B_{1}, D_{1}$ and $F_{1}$ . The volume of the polyhedron bounded on the sides and below by the faces of the prism, and above by the roof $OAB_{1}CD_{1}EF_{1}A$ , does not depend on the choice of the point $O$ . Under what condition will the surface of the polyhedron be least? (The problem of the honeycomb.)

Problem 1565

A plane parallel to opposite edges of a tetrahedron cuts it in a parallelogram. When is the area of this parallelogram greatest?

Problem 1566

Two sides of a parallelogram lie on the sides of a triangle, and a vertex lies on the third side. When is the area of this parallelogram greatest?

Problem 1567

A right circular cone is intersected by a plane in a parabolic segment. The area of such a segment equals $\dfrac {2}{3}$ of the product of the base and the height. Find the greatest area of the segment.

Problem 1568

Two tangents are drawn to a circle. Draw a third tangent so that the triangle bounded by the tangents has the least area.

Problem 1569

Through a point inside a right angle, draw a line so that its segment between the sides of the angle is least.

Problem 1570

Through a point inside a right angle, draw a line so that the perimeter of the resulting triangle is least.

Problem 1571

Find the point on the curve $xy^{2} = a^{2}(a-x)$ for which the subtangent is maximal.

Problem 1572

At what point of the curve $x^{n}y = a^{n+1}$ should the tangent be drawn so that its segment between the coordinate axes is least?

Problem 1573

Find the greatest distance from the origin to the tangent to the curve $x^{n}y = a^{n+1}$ .

Problem 1574

Find the point on the parabola $y^{2} = 2px$ closest to the point $(a, 0)$ . Investigate the result depending on $a$ .

Problem 1575

Find the least length of the segment of the normal to the parabola $x^{2} = 2py$ lying inside the parabola.

Problem 1576

Draw a normal to a parabola that cuts off from it a segment of least area.

Problem 1577

Find the least possible area of the triangle formed by a tangent to the ellipse

\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1

and the coordinate axes.

Problem 1578

Determine the greatest (least) angle between conjugate diameters of the ellipse

\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1.

Problem 1579

Determine the greatest angle between the normal to the ellipse

x = a\cos t, \qquad y = b\sin t

and the diameter drawn to the same point.

Problem 1580

Find the greatest distance from the normal to the ellipse

x = a\cos t, \qquad y = b\sin t

to its center.

Problem 1581

Find the greatest distance of a point of the ellipse

x = a\cos t, \qquad y = b\sin t

from the end of its minor semi-axis. Analyze the various cases ( $a^{2} > b^{2}$ , $b^{2} > a^{2}$ , $a^{2} = b^{2}$ ).

Problem 1582

Inscribe in a semicircle of radius $R$ an ellipse of greatest area ( $s = \pi ab$ ; see the preceding problem).

Problem 1583

Two points of the parabola $y = Ax^{2}$ , having abscissas $a$ and $b$ , are to be joined by a two-segment broken line so that the greatest absolute difference between the corresponding ordinates of the parabola and the broken line is least. Find the position of the intermediate vertex of the broken line and of the points where the broken line meets the parabola. By what factor will this "error" be smaller than the "error" of the chord?

Problem 1584

An arc of the parabola $y = Ax^{2}$ between the points with abscissas $a$ and $b$ is to be replaced by a broken line of $n$ segments with the least "error." Verify that the least "error" equals $\dfrac {h}{2n^{2}}$ , if $h$ is the sagitta of the arc (the error of the chord).

Section 4.5Series, Their Convergence[61/59%]

Problem 1585

By direct division in ascending powers one can establish the identity:

\frac{1}{1-x} = 1+x+x^{2}+\ldots +x^{n-1}+\frac{x^{n}}{1-x}.

Starting from it, show that the series

1+x+x^{2}+\ldots

converges for $\left|x\right| < 1$ and its sum equals $\dfrac {1}{1-x}$ .

Problem 1586

Using the identity $\dfrac {1}{\nu }-\dfrac {1}{\nu +1} = \dfrac {1}{\nu (\nu +1)}$ , prove that the series, whose convergence is to be established directly,

\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\ldots

converges and has sum equal to unity.

Problem 1587

Prove the equality

\frac{1}{1 \cdot 3}+\frac{1}{3 \cdot 5}+\frac{1}{5 \cdot 7}+\ldots = \frac{1}{2}.

Problem 1588

Prove the equality

\frac{1}{1 \cdot 2 \cdot 3}+\frac{1}{2 \cdot 3 \cdot 4}+\frac{1}{3 \cdot 4 \cdot 5}+\ldots = \frac{1}{4}.

Problem 1589

Using the hint of problem 1586 ( $\dfrac {1}{\nu }-\dfrac {1}{\nu +1} = \dfrac {1}{\nu (\nu +1)}$ ), find the sum

\frac{1}{1 \cdot 4}+\frac{1}{2 \cdot 5}+\frac{1}{3 \cdot 6}+\frac{1}{4 \cdot 7}+\frac{1}{5 \cdot 8}+\ldots

Problem 1590

Prove the equality

\sum _{n=1}^{\infty }(v_{n}-v_{n+1}) = v_{1}-v,

where it is assumed that $\lim_{n \to \infty } v_{n} = v$ .

Problem 1591

Having established the equality

\frac{m}{(m+1)(m+2)}+\frac{m}{(m+2)(m+3)}+\frac{m}{(m+3)(m+4)}+\ldots = \frac{m}{m+1},

prove that in this case, as $m \to \infty$ , the limit of the sum of the series is not equal to the sum of the limits of the terms of the series, namely: $\lim_{m \to \infty }(u_{1}+u_{2}+u_{3}+\ldots ) = 1$ , while

\lim u_{1}+\lim u_{2}+\lim u_{3}+\ldots = 0+0+0+\ldots = 0.

Problem 1592

Investigate the convergence of the series for various values of the variable $x$ :

x+\frac{x^{2}}{2}+\frac{x^{3}}{3}+\frac{x^{4}}{4}+\ldots

Problem 1593

Investigate the convergence of the series for various values of the variable $x$ :

x+x^{4}+x^{16}+x^{25}+\ldots

Problem 1594

Investigate the convergence of the series for various values of the variable $x$ :

1-\frac{x}{3}+\frac{x^{2}}{5}-\frac{x^{3}}{7}+\ldots

Problem 1595

Investigate the convergence of the series for various values of the variable $x$ :

\frac{\sin x}{1 \cdot 2}+\frac{\sin 2x}{2 \cdot 3}+\frac{\sin 3x}{3 \cdot 4}+\ldots

Problem 1596

Investigate the convergence of the series for various values of the variable $x$ :

x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\ldots

Problem 1597

Investigate the convergence of the series for various values of the variable $x$ :

\frac{1}{x(x+1)}+\frac{1}{(x+1)(x+2)}+\frac{1}{(x+2)(x+3)}+\ldots

Problem 1598

Investigate the convergence of the series for various values of the variable $x$ :

\frac{1}{x^{2}}+\frac{1}{x^{2}+1^{2}}+\frac{1}{x^{2}+2^{2}}+\ldots

Problem 1599

Investigate the convergence of the series:

1+x+\frac{x^{2}}{1 \cdot 2}+\frac{x^{3}}{1 \cdot 2 \cdot 3}+\ldots

Problem 1600

Investigate the convergence of the series:

1+x+1 \cdot 2x^{2}+1 \cdot 2 \cdot 3x^{3}+\ldots

Problem 1601

Investigate the convergence of the series:

x+2x^{2}+x^{3}+4x^{4}+5x^{5}+\ldots

Problem 1602

Investigate the convergence of the series:

x+\frac{2x^{2}}{3}+\frac{2^{2}x^{3}}{5}+\frac{2^{3}x^{4}}{7}+\ldots

Problem 1603

Investigate the convergence of the series:

x+\frac{1}{2^{2}}x^{2}+\frac{1 \cdot 2}{3^{3}}x^{3}+\frac{1 \cdot 2 \cdot 3}{4^{4}}x^{4}+\ldots

Problem 1604

Investigate the convergence of the series:

x+\frac{1 \cdot 2}{1 \cdot 3}x^{2}+\frac{1 \cdot 2 \cdot 3}{1 \cdot 3 \cdot 5}x^{3}+\frac{1 \cdot 2 \cdot 3 \cdot 4}{1 \cdot 3 \cdot 5 \cdot 7}x^{4}+\ldots

Problem 1605

Investigate the convergence of the series:

\frac{x-1}{2}+\frac{1}{\sqrt{2}}\left(\frac{x-1}{2}\right)^{2}+\frac{1}{\sqrt{3}}\left(\frac{x-1}{2}\right)^{3}+\ldots

Problem 1606

Investigate the convergence of the series:

1+\frac{1}{2x\sqrt{2}}+\frac{1}{4x^{2}\sqrt{3}}+\frac{1}{8x^{3}\sqrt{4}}+\frac{1}{16x^{4}\sqrt{5}}+\ldots

Problem 1607

Investigate the convergence of the series for $a > 0$ , $b > 0$ :

\frac{x-3}{a}+\frac{(x-3)^{2}}{a+b}+\frac{(x-3)^{3}}{a+2b}+\frac{(x-3)^{4}}{a+3b}+\ldots

Problem 1608

Investigate the convergence of the series:

\frac{x}{x+2}-\frac{1}{3}\left(\frac{x}{x+2}\right)^{3}+\frac{1}{5}\left(\frac{x}{x+2}\right)^{5}-\ldots

Problem 1609

Investigate the convergence of the series:

\frac{1+x}{x+\frac{1}{x}}+\frac{5+x}{2\left(x+\frac{1}{x}\right)^{2}}+\frac{25+x}{4\left(x+\frac{1}{x}\right)^{3}}+\frac{125+x}{8\left(x+\frac{1}{x}\right)^{4}}+\ldots

Problem 1610

Investigate the convergence of the series:

\tg x-\frac{\tg ^{2}x}{5}+\frac{\tg ^{3}x}{9}-\frac{\tg ^{4}x}{13}+\ldots

Problem 1611

Investigate the convergence of the series:

1+\frac{2\sin x}{3}+\frac{4\sin ^{2}x}{5}+\frac{8\sin ^{3}x}{7}+\frac{16\sin ^{4}x}{9}+\ldots

Problem 1612

Investigate the convergence of the series:

x+\frac{x^{4}}{1 \cdot 2}+\frac{x^{9}}{1 \cdot 2 \cdot 3}+\ldots = \sum _{n=1}^{\infty }\frac{x^{n^{2}}}{n!}.

Problem 1613

Investigate the convergence of the series:

\frac{1}{e^{x}-1}+\frac{2}{e^{2x}-1}+\frac{3}{e^{3x}-1}+\ldots

Problem 1614

Investigate the convergence of the series $\sum_{n=1}^{\infty } u_{n}$ , if $u_{1}=1$ , and $\dfrac {u_{n+1}}{u_{n}} = \dfrac {3}{4}+\dfrac {(-1)^{n}}{2}$ .

Problem 1615

The series $u_{1}+u_{2}+u_{3}+\ldots$ is formed by the following rule: $u_{n} = \dfrac {1}{2^{m\sigma }}$ , where $2^{m} \leq n < 2^{m+1}$ , and $\sigma$ is a constant number. Prove that for $\sigma > 1$ the series $u_{1}+u_{2}+u_{3}+\ldots$ converges, and its sum equals $\dfrac {2^{\sigma }}{2^{\sigma }-2}$ .

Problem 1616

Using the preceding result (problem 1615), prove that for $\sigma > 1$ the series $1+\dfrac {1}{2^{\sigma }}+\dfrac {1}{3^{\sigma }}+\dfrac {1}{4^{\sigma }}+\ldots$ converges.

Problem 1617

The terms of a series are formed by the rule:

u_{n} = \frac{1}{2^{m\sigma }},

where $2^{m-1} < n \leq 2^{m}$ , and $\sigma$ is a constant number. Prove that for $0 < \sigma < 1$ the series $u_{1}+u_{2}+u_{3}+\ldots$ diverges.

Problem 1618

Prove that the series $1+\dfrac {1}{2^{\sigma }}+\dfrac {1}{3^{\sigma }}+\dfrac {1}{4^{\sigma }}+\ldots$ diverges for $0 < \sigma \leq 1$ .

Problem 1619

Establish that the series $1+\dfrac {1}{\sqrt{2}}+\dfrac {1}{\sqrt{3}}+\dfrac {1}{\sqrt{4}}+\ldots$ is divergent, relying on the inequality

\frac{1}{\sqrt{n}} > \frac{2}{\sqrt{n}+\sqrt{n+1}} = 2\left(\sqrt{n+1}-\sqrt{n}\right).

Problem 1620

Investigate the convergence of the series:

1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\ldots

Problem 1621

Investigate the convergence of the series:

1+\frac{1}{2} \cdot \frac{1}{2}+\frac{1 \cdot 3}{2 \cdot 4} \cdot \frac{1}{3}+\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \cdot \frac{1}{4}+\ldots

Problem 1622

Investigate the convergence of the series:

1+\frac{2}{1} \cdot \frac{1}{2}+\frac{2 \cdot 4}{1 \cdot 3} \cdot \frac{1}{3}+\frac{2 \cdot 4 \cdot 6}{1 \cdot 3 \cdot 5} \cdot \frac{1}{4}+\ldots

Problem 1623

Investigate the convergence of the series for $m > 0$ :

1-\frac{m}{1}+\frac{m(m-1)}{1 \cdot 2}-\frac{m(m-1)(m-2)}{1 \cdot 2 \cdot 3}+\ldots

Problem 1624

(a)(b)

Investigate the convergence of the series:

(a)

$1-\dfrac {1}{3}+\dfrac {1}{5}-\dfrac {1}{6}+\ldots$ ;

(b)

$1-\dfrac {1}{2^{\sigma }}+\dfrac {1}{3^{\sigma }}-\dfrac {1}{4^{\sigma }}+\ldots$ ; $\sigma > 0$ .

Problem 1625

Investigate the convergence of the series for $m > -1$ :

1+\frac{m}{1}+\frac{m(m-1)}{1 \cdot 2}+\frac{m(m-1)(m-2)}{1 \cdot 2 \cdot 3}+\ldots

Problem 1626

Investigate the convergence of the series:

1-\frac{2}{1} \cdot \frac{1}{2}+\frac{2 \cdot 4}{1 \cdot 3} \cdot \frac{1}{3}-\frac{2 \cdot 4 \cdot 6}{1 \cdot 3 \cdot 5} \cdot \frac{1}{4}+\ldots

Problem 1627

Investigate the convergence of the series:

\sin x-\sin \sin x+\sin \sin \sin x-\ldots

Problem 1628

Investigate the convergence of the series:

\frac{\sin x}{1}+\frac{\sin 2x}{2}+\frac{\sin 3x}{3}+\frac{\sin 4x}{4}+\ldots

Problem 1629

Investigate the convergence of the series:

\cos x+\frac{\cos 3x}{3}+\frac{\cos 5x}{5}+\frac{\cos 7x}{7}+\ldots

Problem 1630

Investigate the convergence of the series:

1+\frac{1}{2}-\frac{2}{3}+\frac{1}{4}+\frac{1}{5}-\frac{2}{6}+\frac{1}{7}+\frac{1}{8}-\frac{2}{9}+\ldots

Problem 1631

Investigate the convergence of the series:

1+\frac{1}{2}-\frac{1}{3}-\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\frac{1}{7}-\frac{1}{8}+\ldots

Problem 1632

Using the fundamental criterion of convergence, prove that the sum of an absolutely convergent series does not change if the same terms of the series are arranged in any other order.

Problem 1633

Show that the sum of a non-absolutely convergent series does not change if the terms of the series are rearranged so that none of them is moved from its place by more than $m$ places, where $m$ is any given number.

Problem 1634

Prove the equality

1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\ldots = 2\left(1-\frac{1}{2}-\frac{1}{4}+\frac{1}{3}-\frac{1}{6}-\frac{1}{8}+\frac{1}{5}-\frac{1}{10}-\frac{1}{12}+\ldots \right),

thereby showing that for non-absolutely convergent series the sum depends essentially on the order of the terms.

Problem 1635

Using the equalities

s = 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\ldots = 1-\left(\frac{1}{2}-\frac{1}{3}\right)-\left(\frac{1}{4}-\frac{1}{5}\right)-\ldots ,

s_{1} = 1-\frac{1}{2}+\frac{1}{3}+\frac{1}{5}-\frac{1}{4}+\frac{1}{7}+\frac{1}{9}-\frac{1}{6}+\ldots = 1-\frac{1}{2}+\left(\frac{1}{3}+\frac{1}{5}-\frac{1}{4}\right)+\left(\frac{1}{7}+\frac{1}{9}-\frac{1}{6}\right)+\ldots ,

prove that $s < 1$ , while $s_{1} > 1$ , and hence $s_{1} \neq s$ .

Problem 1636

Prove that the series $1-\dfrac {1}{\sqrt{2}}+\dfrac {1}{\sqrt{3}}-\dfrac {1}{\sqrt{4}}+\dfrac {1}{\sqrt{5}}-\dfrac {1}{\sqrt{6}}+\ldots$ is convergent, while the series $1+\dfrac {1}{\sqrt{3}}-\dfrac {1}{\sqrt{2}}+\dfrac {1}{\sqrt{5}}+\dfrac {1}{\sqrt{7}}-\dfrac {1}{\sqrt{4}}+\dfrac {1}{\sqrt{9}}+\dfrac {1}{\sqrt{11}}-\dfrac {1}{\sqrt{6}}+\ldots$ , obtained from the first by rearranging the terms, is divergent.

Problem 1637

Prove that the series

x(1-x)+x^{2}(1-x)+x^{3}(1-x)+\ldots

converges on the segment $0 \leq x \leq 1$ , but not uniformly.

Problem 1638

Prove that the series

1+x+\frac{x^{2}}{1 \cdot 2}+\frac{x^{3}}{1 \cdot 2 \cdot 3}+\ldots

converges for every $x$ , but not uniformly. The same series converges uniformly on any finite segment $a \leq x \leq b$ .

Problem 1639

Prove that the series

\sum _{n=1}^{\infty } \frac{1}{x^{2}+n^{2}}

converges uniformly on the whole real axis $-\infty < x < \infty$ .

Problem 1640

Prove that the series

\sin x+\frac{\sin 2x}{2^{3}}+\frac{\sin 3x}{3^{3}}+\ldots

can be differentiated term by term.

Problem 1641

Prove that the series

\frac{\cos x}{1 \cdot 2}+\frac{\cos 2x}{2 \cdot 3}+\frac{\cos 3x}{3 \cdot 4}+\ldots

represents a continuous function of $x$ .

Problem 1642

Using Abel's identity, prove that the series

a_{1}\sin x+a_{2}\sin 2x+a_{3}\sin 3x+\ldots ,

a_{1}\cos x+a_{2}\cos 2x+a_{3}\cos 3x+\ldots

when the conditions

a_{1} \geq a_{2} \geq a_{3} \geq a_{4} \geq \ldots , \quad \lim _{n \to \infty } a_{n} = 0

hold, converge uniformly on every segment containing neither inside it nor at its endpoints the values $x = 0, \pm 2\pi , \pm 4\pi , \ldots$ .

Problem 1643

Prove that the sum of the series

s(x) = \sum _{n=1}^{\infty } \frac{\sin 2^{n^{2}}x}{a^{n^{2}}}, \quad a > 1,

represents a continuous function.

Problem 1644

Prove that for $a > 2$ the sum of the series of problem 1643, $s(x) = \sum_{n=1}^{\infty } \dfrac {\sin 2^{n^{2}}x}{a^{n^{2}}}$ , has derivative:

s'(x) = \sum _{n=1}^{\infty } \left(\frac{2}{a}\right)^{n^{2}}\cos 2^{n^{2}}x.

Problem 1645

Prove that for $1 < a < 2$ the function $s(x)$ of problem 1644 has no derivative (while nevertheless being continuous).

Section 4.6Expansion into Series[43/0%]

Problem 1646

(a)(b)

By direct division, arranging the polynomials in ascending powers of the variable $x$ , prove the series expansions:

(a)

$\dfrac {1}{a-bx} = \dfrac {1}{a}+\dfrac {bx}{a^{2}}+\dfrac {b^{2}x^{2}}{a^{3}}+\ldots$ ;

(b)

$\dfrac {1}{a+bx} = \dfrac {1}{a}-\dfrac {bx}{a^{2}}+\dfrac {b^{2}x^{2}}{a^{3}}-\ldots$ , $\left(\left|x\right| < \left|\dfrac {a}{b}\right|\right)$ .

Problem 1647

Using the identity $\dfrac {5-2x}{6-5x+x^{2}} = \dfrac {1}{2-x}+\dfrac {1}{3-x}$ , prove the series expansion:

\frac{5-2x}{6-5x+x^{2}} = \frac{5}{6}+\frac{13}{36}x+\frac{35}{216}x^{2}+\ldots +\left(\frac{1}{2^{n}}+\frac{1}{3^{n}}\right)x^{n-1}+\ldots , \quad \left|x\right| < 2.

Problem 1648

Assuming the series expansion

\frac{1}{1-x-x^{2}} = a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\ldots

to be legitimate, and multiplying both sides of the equality by $1-x-x^{2}$ , prove that $a_{0}=1$ , $a_{1}=1$ , $a_{2}=2$ , $a_{3}=3$ , $a_{4}=5, \ldots$ and in general that $a_{0}, a_{1}, a_{2}, \ldots$ are the Fibonacci numbers, defined by the equalities $a_{n}=a_{n-1}+a_{n-2}$ for $n \geq 2$ , $a_{0}=1$ , $a_{1}=1$ .

Problem 1649

Using the identity

\frac{1}{1-x-x^{2}} = \frac{1}{\sqrt{5}}\left(\frac{1}{\frac{\sqrt{5}+1}{2}+x}+\frac{1}{\frac{\sqrt{5}-1}{2}-x}\right),

find a formula giving the general term $a_{n}$ of the Fibonacci series.

Problem 1650

Prove that in the series expansion of the fraction

\frac{a+bx+cx^{2}}{A+Bx+Cx^{2}+Dx^{3}} = a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\ldots ,

which holds for $A \neq 0$ and for sufficiently small values of $\left|x\right|$ , there exists the relation between the coefficients $a_{n}$ : $Aa_{n}+Ba_{n-1}+Ca_{n-2}+Da_{n-3}=0$ ; $n > 2$ .

Problem 1651

Expand the polynomial $x^{4}-5x^{3}+5x^{2}+x+2$ in powers of $x-2$ .

Problem 1652

Expand the polynomial $x^{5}+2x^{4}-x^{2}+x+1$ in powers of $x+1$ .

Problem 1653

Prove the equalities:

\sin (a+x) = \sin a \cos +\cos a \sin x,

\cos (a+x) = \cos a \cos x-\sin a \sin x,

by expanding the left-hand sides in powers of $x$ .

Problem 1654

In the expansion for $\ln (1+x)$ , replace $x$ by $-x$ ; starting from the series for $\ln (1+x)$ and $\ln (1-x)$ , obtain Gregory's series:

\ln \frac{1+x}{1-x} = 2\left(x+\frac{x^{3}}{3}+\frac{x^{5}}{5}+\ldots \right), \quad \left|x\right| < 1.

Problem 1655

Obtain the equality

\ln (N+1)-\ln N = 2\left(\frac{1}{2N+1}+\frac{1}{3(2N+1)^{3}}+\frac{1}{5(2N+1)^{5}}+\ldots \right), \quad N > 0.

Problem 1656

From the series for $e^{x}$ obtain the expansion for $e^{-x}$ , as well as the expansions:

\ch x \equiv \frac{e^{x}+e^{-x}}{2} = 1+\frac{x^{2}}{2!}+\frac{x^{4}}{4!}+\ldots ,

\sh x \equiv \frac{e^{x}-e^{-x}}{2} = x+\frac{x^{3}}{3!}+\frac{x^{5}}{5!}+\ldots ,

valid for every $x$ .

Problem 1657

Using Euler's formula $e^{ix} = \cos x+i\sin x$ , find the sum $1+\cos x+\cos 2x+\ldots +\cos (n-1)x$ .

Problem 1658

Using Euler's formula $e^{ix} = \cos x+i\sin x$ , find the sum $\cos x+\cos 3x+\cos 5x+\ldots +\cos (2n-1)x$ .

Problem 1659

Using Euler's formula $e^{ix} = \cos x+i\sin x$ , find the sum $\sin x+\sin 2x+\sin 3x+\ldots +\sin nx$ .

Problem 1660

Using Euler's formula $e^{ix} = \cos x+i\sin x$ , find the sum $\sin x-\sin 2x+\ldots -\sin 2nx$ .

Problem 1661

Using Euler's formula $e^{ix} = \cos x+i\sin x$ , find the sum $\sin^{2}x-\sin^{2}3x+\sin^{2}5x-\ldots +(-1)^{n-1}\sin^{2}(2n-1)x$ .

Problem 1662

Using Euler's formula $e^{ix} = \cos x+i\sin x$ , find the sum $x\sin a+x^{2}\sin 2a+\ldots +x^{n-1}\sin (n-1)x$ .

Problem 1663

Prove the equality

\sum _{n=0}^{\infty } \frac{x^{4n}}{(4n)!} = \frac{1}{4}(e^{x}+e^{-x}+2\cos x).

Problem 1664

Prove the equality

\sum _{n=0}^{\infty } \frac{x^{3n}}{(3n)!} = \frac{1}{3}\left(e^{x}+2e^{-x/2}\cos \frac{x\sqrt{3}}{2}\right).

Problem 1665

Prove the equality

e^{x}\cos x = \sum _{n=0}^{\infty } \frac{2^{n/2}\cos \frac{\pi n}{4}}{n!}x^{n}.

Here, as elsewhere, it is assumed that $0! = 1$ .

Problem 1666

Prove the equality

\ch x \cdot \cos x = \sum _{n=0}^{\infty } (-1)^{n}\frac{2^{2n}x^{4n}}{(4n)!}.

Problem 1667

Prove the equality

e^{x\ctg a}\cos x = \sum _{n=0}^{\infty } \frac{\cos na}{\sin ^{n}a} \cdot \frac{x^{n}}{n!}.

Problem 1668

Prove the equality

(1+x^{2})^{m/2}\sin (m\arctg x) = \sum _{\nu =0}^{\infty } (-1)^{\nu }\frac{m(m-1)\ldots (m-2\nu )}{(2\nu +1)!}x^{2\nu +1}, \quad \left|x\right| < 1.

Problem 1669

Prove the equality

(1+x^{2})^{-m/2}\cos (m\arctg x) = \sum _{\nu =0}^{\infty } (-1)^{\nu }\frac{m(m+1)\ldots (m+2\nu -1)}{(2\nu )!}x^{2\nu }, \quad \left|x\right| < 1.

Problem 1670

Applying Lagrange's formula to $\varphi (x)-\varphi (\pi )$ , where

\varphi (x) = \frac{x}{2}+\frac{\sin x}{1}+\frac{\sin 2x}{2}+\ldots +\frac{\sin nx}{n}+\frac{\cos \left(n+\frac{1}{2}\right)x}{(2n+1)\sin \frac{x}{2}},

prove that for $0 < x < 2\pi$ the equality

\frac{\sin x}{1}+\frac{\sin 2x}{2}+\ldots +\frac{\sin nx}{n}+\ldots = \frac{\pi -x}{2}.

holds.

Problem 1671

Applying Lagrange's formula to $\varphi (x)-\varphi \left(\dfrac {\pi }{2}\right)$ , where

\varphi (x) = \frac{\sin x}{1}+\frac{\sin 3x}{3}+\ldots +\frac{\sin (2n-1)x}{2n-1}+\frac{\cos 2nx}{4n\sin x},

prove that for $0 < x < \pi$ the equality

\frac{\sin x}{1}+\frac{\sin 3x}{3}+\frac{\sin 5x}{5}+\ldots +\frac{\sin (2n-1)x}{2n-1}+\ldots = \frac{\pi }{4}.

holds.

Problem 1672

Using the binomial theorem, obtain the expansion

\frac{1}{\sqrt{1-x^{2}}} = 1+\frac{1}{2}x^{2}+\frac{1 \cdot 3}{2 \cdot 4}x^{4}+\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}x^{6}+\ldots , \quad \left|x\right| < 1.

Problem 1673

Prove that for $\left|x\right| < 1$ the equalities hold:

\frac{1}{(1-x)^{2}} = 1+2x+3x^{2}+4x^{3}+\ldots = \sum _{n=0}^{\infty } (n+1)x^{n},

\frac{1}{(1-x)^{3}} = 1+3x+6x^{2}+10x^{3}+\ldots = \sum _{n=0}^{\infty } \frac{(n+1)(n+2)}{2}x^{n}.

Problem 1674

Prove the equality:

\frac{1}{\sqrt{a^{2}+x^{2}}} = \frac{1}{a}-\frac{1}{2}\frac{x^{2}}{a^{3}}+\frac{1 \cdot 3}{2 \cdot 4}\frac{x^{4}}{a^{5}}-\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}\frac{x^{6}}{a^{7}}+\ldots , \quad \left|x\right| < a.

Problem 1675

Prove the equality:

(a+x)^{n} = a^{n}+nxa^{n-1}+\frac{n(n-1)}{1 \cdot 2}x^{2}a^{n-2}+\ldots , \quad \left|x\right| < a.

Problem 1676

Expand the function $\sqrt{1+x}$ in powers of $x$ .

Problem 1677

Expand the function $\ln \dfrac {a+x}{a-x}$ in powers of $x$ .

Problem 1678

Expand the function $\ln (1-x+x^{2})$ in powers of $x$ .

Problem 1679

Expand the function $\ln (2-3x+x^{2})$ in powers of $x$ .

Problem 1680

Expand the function $\dfrac {(1+x)^{2}}{x}\ln (1+x)$ in powers of $x$ .

Problem 1681

Prove the equality

\ln (1-2x\cos \varphi +x^{2}) = -2\sum _{n=1}^{\infty } \frac{\cos n\varphi }{n}x^{n}, \quad \left|x\right| < 1.

Problem 1682

Verify that

\frac{1-a\cos x}{1-2a\cos x+a^{2}} = \sum _{n=0}^{\infty } a^{n}\cos nx, \quad \left|a\right| < 1.

Problem 1683

Verify that

\frac{a\sin x}{1-2a\cos x+a^{2}} = \sum _{n=1}^{\infty } a^{n}\sin nx, \quad \left|a\right| < 1.

Problem 1684

By a suitable change of variable, obtain the series expansion:

\ln \frac{x+1}{x-1} = 2\left(\frac{1}{x}+\frac{1}{3x^{3}}+\frac{1}{5x^{5}}+\ldots \right), \quad \left|x\right| > 1.

Problem 1685

By a suitable change of variable, obtain the series expansion:

\ln x = 2\left(\frac{x-1}{x+1}+\frac{1}{3}\left(\frac{x-1}{x+1}\right)^{3}+\frac{1}{5}\left(\frac{x-1}{x+1}\right)^{5}+\ldots \right), \quad x > 0.

Problem 1686

By a suitable change of variable, obtain the series expansion:

\frac{x}{\sqrt{1+x}} = \frac{x}{x+1}+\frac{1}{2}\left(\frac{x}{x+1}\right)^{2}+\frac{1 \cdot 3}{2 \cdot 4}\left(\frac{x}{x+1}\right)^{3}+\ldots , \quad x > -\frac{1}{2}.

Problem 1687

Obtain the series expansion:

\ln \left(x+\sqrt{1+x^{2}}\right) = \sum _{n=0}^{\infty } (-1)^{n}\frac{1 \cdot 3 \cdot \ldots \cdot (2n-1)}{2 \cdot 4 \cdot \ldots \cdot 2n}\frac{x^{2n+1}}{2n+1}, \quad \left|x\right| < 1.

Problem 1688

Obtain the series expansion:

\ln \frac{2\left(1-\sqrt{1-x}\right)}{x} = \frac{1}{2} \cdot \frac{x}{2}+\frac{1 \cdot 3}{2 \cdot 4}\frac{x^{2}}{4}+\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}\frac{x^{3}}{6}+\frac{1 \cdot 3 \cdot 5 \cdot 7}{2 \cdot 4 \cdot 6 \cdot 8}\frac{x^{4}}{8}+\ldots , \quad \left|x\right| < 1.

Section 4.8Evaluation of Indeterminate Forms[39/64%]

Problem 1746

Find the limit $\lim_{x \to 0} \dfrac {\ln (1-x)+x^{2}}{(1+x)^{m}-1+x^{2}}$ .

Problem 1747

Find the limit $\lim_{x \to \pi } \dfrac {\sin 3x}{\tg 5x}$ .

Problem 1748

Find the limit $\lim_{x \to 0} \dfrac {\tg x-1+\cos 3x}{e^{x}-e^{-x}}$ .

Problem 1749

Find the limit $\lim_{x \to \frac{\pi }{4}} \dfrac {\tg x-1}{\sin 4x}$ .

Problem 1750

Find the limit $\lim_{x \to 0} \dfrac {\ln \cos ax}{\ln \cos bx}$ .

Problem 1751

Find the limit $\lim_{x \to 0} \dfrac {e^{x}-e^{-x}-2x}{x-\sin x}$ .

Problem 1752

Find the limit $\lim_{x \to +\infty } \dfrac {x^{3}}{e^{x}}$ .

Problem 1753

Find the limit $\lim_{x \to +\infty } \dfrac {x^{m}}{e^{ax}}$ , $a > 0$ .

Problem 1754

Find the limit $\lim_{x \to 1} (1-x)\tg \dfrac {\pi x}{2}$ .

Problem 1755

Find the limit $\lim_{x \to 0} x^{x}$ .

Problem 1756

Find the limit $\lim_{x \to a} \arcsin (x-a)\ctg (x-a)$ .

Problem 1757

Find the limit $\lim_{x \to 0} \dfrac {e^{-1/x^{2}}}{x^{m}}$ , $m > 0$ .

Problem 1758

Find the limit $\lim_{x \to +\infty } \dfrac {\ln x}{x^{m}}$ , $m > 0$ .

Problem 1759

Applying series expansions, find the limit $\lim_{x \to 0} \dfrac {x-\sin x}{e^{x}-1-x-\frac{x^{2}}{2}}$ .

Problem 1760

Find the limit $\lim_{x \to 0} \dfrac {\ln^{2}(1+x)-\sin^{2}x}{1-e^{-x^{2}}}$ .

Problem 1761

Find the limit $\lim_{x \to \infty } \left[x-x^{2}\ln \left(1+\frac{1}{x}\right)\right]$ .

Problem 1762

Find the limit $\lim_{x \to 0} \left(\dfrac {1}{x^{2}}-\ctg^{2}x\right)$ .

Problem 1763

Find the limit $\lim_{x \to 0} \dfrac {\ln (1+x+x^{2})+\ln (1-x+x^{2})}{x\sin x}$ .

Problem 1764

Find the limit $\lim_{x \to 0} \left(\dfrac {1}{x^{2}}-\dfrac {1}{\sin^{2}x}\right)$ .

Problem 1765

Find the limit $\lim_{x \to 0} \dfrac {xe^{\cos x}}{1-\sin x-\cos x}$ .

Problem 1766

Find the limit $\lim_{x \to 0} \dfrac {x^{3}\sin x}{(1-\cos x)^{2}}$ .

Problem 1767

Find the limit $\lim_{h \to 0} \dfrac {f(x+h)-2f(x)+f(x-h)}{h^{2}}$ .

Problem 1768

Find the limit $\lim_{h \to 0} \dfrac {f(x+3h)-3f(x+2h)+3f(x+h)-f(x)}{h^{3}}$ .

Problem 1769

Find the limit by various methods: $\lim_{x \to \infty } \dfrac {x}{x+\sin x}$ .

Problem 1770

Find the limit by various methods: $\lim_{x \to 1} \ln x \cdot \ln (1-x)$ .

Problem 1771

Find the limit by various methods: $\lim_{x \to 0} \dfrac {e^{x}-1+x^{3}\sin \frac{\pi }{x}}{x}$ .

Problem 1772

Find the limit by various methods: $\lim_{x \to 1} x^{\frac{1}{1-x}}$ .

Problem 1773

Find the limit by various methods: $\lim_{x \to +\infty } \dfrac {\ln (a+be^{x})}{\sqrt{m+nx^{2}}}$ , $b > 0$ , $n > 0$ .

Problem 1774

Find the limit by various methods: $\lim_{x \to \infty } \left(\cos \dfrac {a}{x}\right)^{x^{2}}$ .

Problem 1775

Find the limit by various methods: $\lim_{x \to \frac{\pi }{2}} (\sin x)^{\tg x}$ .

Problem 1776

Find the limit by various methods: $\lim_{x \to 0} (\arcsin x)^{\tg x}$ .

Problem 1777

Find the limit by various methods: $\lim_{x \to \infty } \left(1-x\ln \left(1+\frac{1}{x}\right)\right)$ .

Problem 1778

Find the limit by various methods: $\lim_{x \to 1} \left(\dfrac {\pi }{2}\tg \dfrac {\pi x}{2}-\dfrac {1}{1-x}\right)$ .

Problem 1779

Find the limit by various methods: $\lim_{x \to 0} \dfrac {(1+x)^{1/x}-e}{x}$ .

Problem 1780

Find the limit by various methods: $\lim_{x \to 0} \left(\sqrt{\dfrac {1}{x(x-1)}+\dfrac {1}{x^{2}}}-\dfrac {1}{x}\right)$ .

Problem 1781

Find the limit of the ratio of the area of a circular segment to the area of the triangle formed by the chord and the tangents at the ends of the arc, as the arc of the segment tends to zero.

Problem 1782

The same question for the segment and the triangle formed by the chord and the two chords joining its ends to the midpoint of the arc.

Problem 1783

If $a$ , $b$ , $c$ are the sides of a spherical triangle, and $A$ , $B$ , $C$ the opposite angles, then

\cos c = \cos a \cos b + \sin a \sin b \cos C.

What does this formula become if $a$ , $b$ and $c$ are small?

Problem 1784

A body falls in a medium whose resistance is proportional to the first power of the speed. The distance traveled in $t$ seconds is expressed by the formula

s = \frac{mgt}{a}-\frac{m^{2}g}{a^{2}}\left(1-e^{-\frac{at}{m}}\right),

where $m$ is the mass, $a$ the coefficient of friction. Find approximate formulas for $s$ when $\dfrac {at}{m}$ is large, and also for the case when $\dfrac {at}{m}$ is very small.

Section 4.9Extremal Values of Functions of Several Variables[89/94%]

Problem 1785

Find the extreme values of the function given explicitly: $z = x^{2}+xy+y^{2}-3ax-3by$ .

Problem 1786

Find the extreme values of the function given explicitly: $z = x^{3}y^{2}(a-x-y)$ .

Problem 1787

Find the extreme values of the function given explicitly: $z = x^{4}+y^{4}-2x^{2}+4xy-2y^{2}$ .

Problem 1788

Find the extreme values of the function given explicitly: $z = x^{2}-xy+y^{2}+3x-2y+1$ .

Problem 1789

Find the extreme values of the function given explicitly: $z = x^{2}+xy+y^{2}+\dfrac {a^{3}}{x}+\dfrac {a^{3}}{y}$ , $x > 0$ , $y > 0$ .

Problem 1790

Find the extreme values of the function given explicitly: $z = x^{3}+y^{3}-3axy$ .

Problem 1791

Find the extreme values of the function given explicitly: $z = Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F$ .

Problem 1792

Find the extreme values of the function given explicitly: $z = x^{3}+y^{3}-9xy+27$ for $0 \leqslant x \leqslant a$ , $0 \leqslant y \leqslant a$ ; $a > 3$ .

Problem 1793

Find the extreme values of the function given explicitly: $z = x^{4}+y^{4}-2x^{2}+4xy-2y^{2}$ for $0 \leqslant x \leqslant a$ , $0 \leqslant y \leqslant a$ ; $a > 1$ .

Problem 1794

Find the extreme values of the function given explicitly: $z = e^{-x^{2}-y^{2}}(ax^{2}+by^{2})$ ; $a > 0$ , $b > 0$ .

Problem 1795

Find the extreme values of the function given explicitly: $z = \sqrt{(a-x)(a-y)(x+y-a)}$ .

Problem 1796

Find the extreme values of the function given explicitly: $z = \dfrac {a+bx+cy}{\sqrt{1+x^{2}+y^{2}}}$ .

Problem 1797

Find the extreme values of the function given explicitly: $z = \sin x+\sin y+\sin (x+y)$ ; $0 \leqslant x \leqslant \dfrac {\pi }{2}$ ; $0 \leqslant y \leqslant \dfrac {\pi }{2}$ .

Problem 1798

Find the extreme values of the function given explicitly: $z = \sin x+\sin y+\cos (x+y)$ ; $0 \leqslant x \leqslant \dfrac {3\pi }{2}$ ; $0 \leqslant y \leqslant \dfrac {3\pi }{2}$ .

Problem 1799

Find the extreme values of the function given explicitly: $z = \cos x \cos y \cos (x+y)$ ; $0 \leqslant x \leqslant \pi$ , $0 \leqslant y \leqslant \pi$ .

Problem 1800

Find the extreme values of the function given explicitly: $z = (a\cos x+b\cos y)^{2}+(a\sin x+b\sin y)^{2}$ .

Problem 1801

Find the extreme values of the function given explicitly: $z = \cos x \cos \alpha +\sin x \sin \alpha \cos (y-\beta )$ .

Problem 1802

Find the extreme values of the function given explicitly: $u = xyz(4a-x-y-z)$ .

Problem 1803

Find the extreme values of the function given explicitly: $u = x^{2}+y^{2}+z^{2}-xy+2z+x$ .

Problem 1804

Find the extreme values of the function given explicitly: $u = \dfrac {x^{3}+y^{3}+z^{3}}{xyz}$ ; $x > 0$ , $y > 0$ , $z > 0$ .

Problem 1805

Find the extreme values of the function given explicitly: $u = \dfrac {x}{y+z}+\dfrac {y}{x+z}+\dfrac {z}{x+y}$ ; $x > 0$ , $y > 0$ , $z > 0$ .

Problem 1806

Find the extreme values of the function given explicitly: $u = (ax+by+cz)e^{-x^{2}-y^{2}-z^{2}}$ .

Problem 1807

Between the positive numbers $a$ and $b$ , insert $n$ other numbers $x_{1}, x_{2}, \ldots , x_{n}$ so that the value of the fraction

u = \frac{x_{1}x_{2}\ldots x_{n}}{(a+x_{1})(x_{1}+x_{2})\ldots (x_{n-1}+x_{n})(x_{n}+b)}

attains its greatest value.

Problem 1808

Find the greatest and least values of the function $z$ , given implicitly as a function of $x$ and $y$ by the equation $2x^{2}+2y^{2}+z^{2}+8xz-z+8 = 0$ .

Problem 1809

Find the greatest and least values of the function $z$ , given implicitly as a function of $x$ and $y$ by the equation $2x^{2}+6y^{2}+2z^{2}+8xz-4x-8y+3 = 0$ .

Problem 1810

Find the greatest and least values of the function $z$ , given implicitly as a function of $x$ and $y$ by the equation $6x^{2}+6y^{2}+6z^{2}+4x-8y-8z+5 = 0$ .

Problem 1811

Find the greatest and least values of the function $z$ , given implicitly as a function of $x$ and $y$ by the equation $5x^{2}+5y^{2}+5z^{2}-2xy-2xz-2yz-72 = 0$ .

Problem 1812

Find the greatest and least values of the function $z$ , given implicitly as a function of $x$ and $y$ by the equation $x^{3}y-3xy^{2}+6x+y^{2}+7y+z^{2}-3z-14 = 0$ .

Problem 1813

Find the greatest and least values of the function $z$ , given implicitly as a function of $x$ and $y$ by the equation $x^{4}+y^{4}+z^{4} = 2a^{2}(x^{2}+y^{2}+z^{2})$ .

Problem 1814

Find the greatest and least values of the function $u = x+y$ under the condition $\dfrac {1}{x^{2}}+\dfrac {1}{y^{2}} = \dfrac {1}{a^{2}}$ .

Problem 1815

Find the greatest and least values of the function $u = x^{m}+y^{m}$ under the condition $x+y = 2a$ ; $a > 0$ , $m > 1$ .

Problem 1816

Find the greatest and least values of the function $u = x_{1}^{m}+x_{2}^{m}+\ldots +x_{n}^{m}$ under the condition $x_{1}+x_{2}+\ldots +x_{n} = na$ ; $a > 0$ , $m > 1$ .

Problem 1817

Find the greatest and least values of the function $u = x_{1}^{m}+x_{2}^{m}+\ldots +x_{n}^{m}$ under the condition $x_{1}+x_{2}+\ldots +x_{n} = na$ ; $a > 0$ , $m < 1$ .

Problem 1818

Find the greatest and least values of the function $u = xy$ under the condition $x^{2}+y^{2} = 1$ .

Problem 1819

Find the greatest and least values of the function $u = xyz$ under the condition $x^{2}+y^{2}+z^{2} = 3$ .

Problem 1820

Find the greatest and least values of the function $u = x+y+z$ under the condition $\dfrac {a}{x}+\dfrac {b}{y}+\dfrac {c}{z} = 1$ ; $a > 0$ , $b > 0$ , $c > 0$ .

Problem 1821

Find the greatest and least values of the function $u = x^{2}y^{3}z^{4}$ under the condition $2x+3y+4z = a$ .

Problem 1822

Find the greatest and least values of the function $u = x^{2}+2y^{2}+3z^{2}$ under the conditions $x^{2}+y^{2}+z^{2} = 1$ , $x+2y+3z = 0$ .

Problem 1823

Find the greatest and least values of the function $u = xyz$ under the conditions $x+y+z = 5$ , $xy+yz+xz = 8$ .

Problem 1824

Find the greatest and least values of the function $u = x^{2}+y^{2}+z^{2}$ under the conditions $lx+my+nz = 0$ , $(x^{2}+y^{2}+z^{2})^{2} = a^{2}x^{2}+b^{2}y^{2}+c^{2}z^{2}$ .

Problem 1825

Find the greatest and least values of the function $u = \sin \dfrac {x}{2}\sin \dfrac {y}{2}\sin \dfrac {z}{2}$ under the condition $x+y+z = \pi$ ; $x > 0$ , $y > 0$ , $z > 0$ .

Problem 1826

Find the greatest and least values of the function $u = x-y$ under the condition $\tg x = 3\tg y$ ; $0 \leqslant x \leqslant \dfrac {\pi }{2}$ , $0 \leqslant y \leqslant \dfrac {\pi }{2}$ .

Problem 1827

Find the greatest and least values of the function $u = \dfrac {\alpha_{1}}{x_{1}}+\dfrac {\alpha_{2}}{x_{2}}+\ldots +\dfrac {\alpha_{n}}{x_{n}}$ under the condition $\beta_{1}x_{1}+\beta_{2}x_{2}+\ldots +\beta_{n}x_{n} = p$ , where all $\alpha$ and $\beta$ are positive, as are the numbers $x_{1}, x_{2}, \ldots , x_{n}$ .

Problem 1828

Find the greatest and least values of the function $u = Ax^{2}+By^{2}+Cz^{2}+Dxy+Exz+Fyz$ under the condition $x^{2}+y^{2}+z^{2} = a^{2}$ .

Problem 1829

Find the greatest and least values of the function $u = \dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}$ under the conditions $x^{2}+y^{2}+z^{2} = 1$ , $lx+my+nz = 0$ .

Problem 1830

Hadamard's inequality for the determinant of the third order

u = \begin{vmatrix} a & b & c \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{vmatrix}

has the form

\left|u\right| \leqslant 1, \quad \text{if } a^{2}+b^{2}+c^{2} = 1,\ a_{1}^{2}+b_{1}^{2}+c_{1}^{2} = 1,\ a_{2}^{2}+b_{2}^{2}+c_{2}^{2} = 1.

Prove this inequality.

Problem 1831

Maclaurin's inequalities: if $x_{1}, x_{2}, \ldots , x_{n} \geqslant 0$ and $x_{1}+x_{2}+\ldots +x_{n} = na$ , then

x_{1}x_{2}+x_{1}x_{3}+\ldots +x_{n-1}x_{n} \leqslant \frac{n(n-1)}{1\cdot 2}a^{2},

x_{1}x_{2}x_{3}+x_{1}x_{2}x_{4}+\ldots +x_{n-2}x_{n-1}x_{n} \leqslant \frac{n(n-1)(n-2)}{1\cdot 2\cdot 3}a^{3},

and so on. Prove these inequalities.

Problem 1832

Huygens's problem (see problem 1807) can be reduced to proving the inequality:

(1+u_{1})(1+u_{2})\ldots (1+u_{n}) \geqslant (1+q)^{n}; \quad u_{k} > 0,

if $u_{1}u_{2}\ldots u_{n} = q_{n}$ . Prove this inequality.

Problem 1833

Of all triangles with the same perimeter $2p$ , find the one with the greatest area.

Problem 1834

Find the greatest area of a triangle with given base $a$ and angle $A$ at the apex.

Problem 1835

Divide a given triangle into two parts of equal area by a segment of least length.

Problem 1836

Prove that in a triangle the radius of the inscribed circle is no greater than half the radius of the circumscribed circle.

Problem 1837

Inscribe in a given triangle a triangle of least perimeter.

Problem 1838

Inscribe in a given square a quadrilateral of least perimeter.

Problem 1839

Inside a quadrilateral, find the point for which the sum of the squares of the distances to the vertices is least.

Problem 1840

Find the point for which the sum of the squares of the distances to given points is least.

Problem 1841

Find the greatest area of a quadrilateral with given sides $a$ , $b$ , $c$ , $d$ .

Problem 1842

Find the polygon with the greatest area having given sides $a_{1}, a_{2}, \ldots$ .

Problem 1843

Find the greatest area of a polygon with $n$ sides inscribed in a circle of radius $a$ .

Problem 1844

Find the least area of a polygon with $n$ sides circumscribed about a circle of radius $a$ .

Problem 1845

Inscribe in a given circle a triangle for which the sum of the squares of the sides is greatest.

Problem 1846

Inscribe in a given circle a quadrilateral of greatest area with given angle $\alpha$ .

Problem 1847

On the plane of a triangle, find the point for which the sum of the squares of the distances to the sides of the triangle is least.

Problem 1848

Of all planes passing through a given point, find the one most distant from the origin.

Problem 1849

On the ellipsoid

\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} = 1

find the point most distant from the origin.

Problem 1850

Find the greatest volume of a parallelepiped with given sum of its edges $12a$ .

Problem 1851

Find the greatest volume of a rectangular parallelepiped with given total surface $6a^{2}$ .

Problem 1852

Circumscribe about a rectangular parallelepiped with edges $2a$ , $2b$ and $2c$ the ellipsoid of least volume.

Problem 1853

Through the point $(a, b, c)$ , draw a plane forming with the coordinate planes a tetrahedron of least volume.

Problem 1854

Determine the dimensions of a rectangular box, open at the top, so that for given volume $V$ and wall thickness $h$ the least amount of material is used.

Problem 1855

Determine the dimensions of a cylindrical vessel with given surface and greatest volume.

Problem 1856

Inscribe in a given cone a rectangular parallelepiped of greatest volume.

Problem 1857

Which of the cones with given lateral surface $S$ has the greatest volume?

Problem 1858

Find the vessel of greatest volume having the shape of a frustum of a cone with given base and slant height: $l = 4r$ .

Problem 1859

Two points are given on an ellipse. Where on the same ellipse should a third be placed so that the resulting triangle has greatest area?

Problem 1860

Find the shortest distance from the point $(p, 4p)$ to the parabola $y^{2} = 2px$ .

Problem 1861

Inscribe in a given isosceles triangle a parabolic segment with common axis and greatest area. The area of a parabolic segment equals $\dfrac {2}{3}$ of the product of the base and the height.

Problem 1862

Inscribe in a given ellipse an isosceles triangle of greatest area with base parallel to an axis.

Problem 1863

On an ellipse, find the point closest to the given point $(m, 0)$ of the major axis.

Problem 1864

Circumscribe about an ellipse a triangle of least area with base parallel to an axis.

Problem 1865

Find the normal to an ellipse most distant from the center.

Problem 1866

Find the normal to the ellipse with semi-axes $a$ and $b$ whose segment inside the ellipse has least length.

Problem 1867

Draw to an ellipse a tangent whose segment between the axes has least length.

Problem 1868

Find the area $S$ of the ellipse obtained by cutting the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}} = 1$ with the plane $lx+my+nz = 0$ .

Problem 1869

Draw to the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}} = 1$ the tangent plane with least sum of intercepts on the axes.

Problem 1870

Prove that of the chords parallel to the axis $Oz$ and lying between the plane $z = ax+by+c$ and the elliptic paraboloid $z = Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F$ , the maximal one passes through the center of the plane section.

Problem 1871

In the segment of the elliptic paraboloid $z = \dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}$ cut off by the plane $z = h$ , inscribe a rectangular parallelepiped of greatest volume.

Problem 1872

To the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}} = 1$ , draw a tangent plane such that the centroid of the triangle cut off on this plane by the coordinate planes is least distant from the origin.

Problem 1873

To the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}} = 1$ , draw a tangent plane forming, with its planes of symmetry, a tetrahedron of least volume.