Math Problems Vault

Problem 1298

Examine the concavity or convexity of the curve

y=1+\sqrt[3]{x}

at the points $A(-1,0), B(1,2)$ and $C(0,0)$ .

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Problem 1299

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=3x^{2}-x^{3} .

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Problem 1300

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=\frac{a^{3}}{a^{2}+x^{2}} \ (a>0) .

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Problem 1301

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=x+x^{\frac{5}{3}} .

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Problem 1302

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=\sqrt{1+x^{2}} .

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Problem 1303

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=x+\sin x .

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Problem 1304

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=e^{-x^{2}} .

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Problem 1305

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=\ln (1+x^{2}) .

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Problem 1306

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=x\sin (\ln x) \ (x>0) .

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Problem 1307

Find the regions of concavity or convexity and the inflection points of the graph of the following function:

y=x^{x} \ (x>0) .

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Problem 1308

Prove that the curve

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

has three inflection points lying on the same straight line. Sketch the graph of this function.

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Problem 1309

How should the parameter $h$ be chosen so that the "probability curve"

y=\frac{h}{\sqrt{\pi }}e^{-h^{2}x^{2}} \ (h>0)

has inflection points $x=\pm \sigma$ ?

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Problem 1310

Examine the concavity or convexity of the cycloid

x=a(t-\sin t), y=a(1-\cos t) \ (a>0)

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Problem 1311

Let the function $f(x)$ be twice differentiable on the interval $a \leqslant x<+\infty$ , and:

$f(a)=A>0$ ;
$f'(a)<0$ ;
when $x>a$ , $f''(x) \leqslant 0$ . Prove: the equation $f(x)=0$ has one and only one real root in the interval $(a,+\infty )$ .

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Problem 1312

(a)(b)

If for any two points $x_{1}$ and $x_{2}$ in the interval $(a,b)$ and any two numbers $\lambda_{1}$ and $\lambda_{2} \ (\lambda_{1}>0, \lambda_{2}>0, \lambda_{1}+\lambda_{2}=1)$ we have the inequality:

f(\lambda _{1}x_{1}+\lambda _{2}x_{2})<\lambda _{1}f(x_{1})+\lambda _{2}f(x_{2})

(or correspondingly, the reverse inequality $f(\lambda_{1}x_{1}+\lambda_{2}x_{2})>\lambda_{1}f(x_{1})+\lambda_{2}f(x_{2})$ ), then the function $f(x)$ is said to be concave (convex) on the interval $(a,b)$ .

Prove that the function $f(x)$ : (1) if $f''(x)>0$ for $a<x<b$ , then it is concave on $(a,b)$ ; (2) if $f''(x)<0$ for $a<x<b$ , then it is convex on $(a,b)$ .

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(a)

(b)

Problem 1313

Prove that the functions

x^{n} \ (n>1), e^{x}, x\ln x

are concave on the interval $(0,+\infty )$ ; while the functions

x^{n} \ (0<n<1), \ln x .

are convex on the interval $(0,+\infty )$ .

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Problem 1314

(a)(b)(c)

Prove the following inequalities, and explain their geometric meaning:

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(a)

(b)

$\dfrac {e^{x}+e^{y}}{2}>e^{\frac{x+y}{2}} \ (x \neq y)$ ;

(c)

$x\ln x+y\ln y>(x+y)\ln \dfrac {x+y}{2}$

(x>0, y>0) .

Problem 1314.1

$\dfrac {1}{2}(x^{n}+y^{n})>\left(\dfrac {x+y}{2}\right)^{n}, \ (x>0, y>0, x \neq y, n>1)$ ;

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Problem 1315

Prove that a bounded convex function is continuous everywhere, and has left-hand and right-hand derivatives.

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Problem 1316

Let the function $f(x)$ be twice differentiable on the interval $(a,b)$ , with $f''(\xi ) \neq 0$ , where $a<\xi <b$ . Prove that two values $x_{1}$ and $x_{2}$ can be found in the interval $(a,b)$ satisfying

\frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}}=f'(\xi ) .

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Problem 1317

Prove that if the function $f(x)$ is twice differentiable on the infinite interval $(x_{0},+\infty )$ , and

\lim _{x \to x_{0}+0}f(x)=0, \quad \lim _{x \to +\infty }f(x)=0 ,

then there is at least one point $\xi$ in the interval $(x_{0},+\infty )$ satisfying

f''(\xi )=0 .

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