Math Problems Vault

Problem 1268

Find the intervals of strict monotonicity (increase or decrease) of the following function: $y=2+x-x^{2}$ .

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Problem 1269

Find the intervals of strict monotonicity (increase or decrease) of the following function: $y=3x-x^{3}$ .

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Problem 1270

Find the intervals of strict monotonicity (increase or decrease) of the following function: $y=\dfrac {2x}{1+x^{2}}$ .

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Problem 1271

Find the intervals of strict monotonicity (increase or decrease) of the following function: $y=\dfrac {\sqrt{x}}{x+100} \ (x \geqslant 0)$ .

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Problem 1272

Find the intervals of strict monotonicity (increase or decrease) of the following function: $y=x+\sin x$ .

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Problem 1273

Find the intervals of strict monotonicity (increase or decrease) of the following function: $y=x+\left|\sin 2x\right|$ .

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Problem 1274

Find the intervals of strict monotonicity (increase or decrease) of the following function: $y=\cos \dfrac {\pi }{x}$ .

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Problem 1275

Find the intervals of strict monotonicity (increase or decrease) of the following function: $y=\dfrac {x^{2}}{2^{x}}$ .

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Problem 1276

Find the intervals of strict monotonicity (increase or decrease) of the following function: $y=x^{n}e^{-x} \ (n>0, x \geqslant 0)$ .

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Problem 1277

Find the intervals of strict monotonicity (increase or decrease) of the following function: $y=x^{2}-\ln x^{2}$ .

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Problem 1278

If $x>0, f(x)=x\left(\sqrt{\dfrac {3}{2}}+\sin \ln x\right)$ and $f(0)=0$ .

Discuss the monotonicity of the function $f(x)$ .

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Problem 1279

Prove that for a regular $n$ -gon inscribed in a circle, as the number of sides $n$ increases, its perimeter $p_{n}$ increases, while the perimeter $P_{n}$ of the regular $n$ -gon circumscribed about this circle decreases. Use this to prove that as $n \to \infty$ , $p_{n}$ and $P_{n}$ have the same limit.

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Problem 1280

Prove that the function

\left(1+\frac{1}{x}\right)^{x}

is increasing on the intervals $(-\infty ,-1)$ and $(0,+\infty )$ .

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Problem 1281

Prove that the polynomial

P(x)=a_{0}+a_{1}x+\cdots +a_{n}x^{n} \ (n \geqslant 1, a_{n} \neq 0)

is monotonic (in the strict sense!) on the intervals $(-\infty ,-x_{0})$ and $(x_{0},+\infty )$ , where $x_{0}$ is a sufficiently large positive number.

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Problem 1282

Prove that the rational function

R(x)=\frac{a_{0}+a_{1}x+\cdots +a_{n}x^{n}}{b_{0}+b_{1}x+\cdots +b_{m}x^{m}}

(m+n \geqslant 1, m \neq n, a_{n}b_{m} \neq 0)

is monotonic (in the strict sense!) on the intervals $(-\infty ,-x_{0})$ and $(x_{0},+\infty )$ , where $x_{0}$ is a sufficiently large positive number.

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Problem 1283

Must the derivative of a monotonic function also be monotonic?

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Problem 1284

Prove that if $\varphi (x)$ is a monotonically increasing differentiable function, and when

x \geqslant x_{0}, \left|f'(x)\right| \leqslant \varphi (x),

then when $x \geqslant x_{0}$ , $\left|f(x)-f(x_{0})\right| \leqslant \varphi (x)-\varphi (x_{0})$ .

Give a geometric interpretation of this fact.

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Problem 1285

Suppose the function $f(x)$ is continuous on the interval $a \leqslant x<+\infty$ , and when $x>a$ , $f'(x)>k>0$ , where $k$ is a constant. Prove: if $f(a)<0$ , then on the interval $\left(a,a-\dfrac {f(a)}{k}\right)$ the equation $f(x)=0$ has one and only one real root.

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Problem 1286

If in some neighborhood $\left|x-x_{0}\right|<\delta$ , the increment of the function $\Delta f(x_{0})=f(x)-f(x_{0})$ has the same sign as the increment of the independent variable $\Delta x_{0}=x-x_{0}$ , the function $f(x)$ is said to be increasing at $x_{0}$ .

Prove: if the function $f(x) \ (a<x<b)$ is increasing at every point of the finite or infinite interval $(a,b)$ , then it is an increasing function on this interval.

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Problem 1287

Prove that the function

f(x)=x+x^{2}\sin \frac{1}{x}, \text{ if } x \neq 0, \text{ and } f(0)=0,

is increasing at the point $x=0$ , but is not increasing on any interval $(-\varepsilon ,\varepsilon )$ containing this point, where $\varepsilon >0$ is an arbitrarily small number. Sketch this function.

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Problem 1288

Prove the theorem: suppose (1) the functions $\varphi (x)$ and $\psi (x)$ are $n$ times differentiable; (2) $\varphi^{(k)}(x_{0})=\psi^{(k)}(x_{0}) \ (k=0,1,2,\cdots ,n-1)$ ; (3) when $x>x_{0}$ , $\varphi^{(n)}(x)>\psi^{(n)}(x)$ ; then when $x>x_{0}$ the inequality

\varphi (x)>\psi (x)

holds.

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Problem 1289

Prove the following inequalities:

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(a)

When $x \neq 0$ , $e^{x}>1+x$ ;

(b)

When $x>0$ , $x-\dfrac {x^{2}}{2}<\ln (1+x)<x$ ;

(c)

When $x>0$ , $x-\dfrac {x^{3}}{6}<\sin x<x$ ;

(d)

When $0<x<\dfrac {\pi }{2}$ , $\tg x>x+\dfrac {x^{3}}{3}$ ;

(e)

When $x>0, y>0$ and $0<\alpha <\beta$ ,

(x^{\alpha }+y^{\alpha })^{\frac{1}{\alpha }}>(x^{\beta }+y^{\beta })^{\frac{1}{\beta }} .

Give a geometric interpretation of inequalities (a)--(4).

Problem 1290

Prove the inequality

\frac{2}{\pi }x<\sin x<x,

holds when $0<x<\dfrac {\pi }{2}$ .

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Problem 1291

Prove that when $x>0$ the inequality

\left(1+\frac{1}{x}\right)^{x}<e<\left(1+\frac{1}{x}\right)^{x+1}

holds.

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Problem 1292

An arithmetic progression and a geometric progression have the same number of terms and the same first and last terms, and all their terms are positive. Prove that the sum of the terms of the arithmetic progression is greater than or equal to the sum of the terms of the geometric progression.

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Problem 1293

Using the inequality

\sum _{k=1}^{n}(a_{k}x+b_{k})^{2} \geqslant 0 ,

where $x,a_{k},b_{k} \ (k=1,2,\cdots ,n)$ are real numbers, prove the Cauchy--Bunyakovsky inequality

\left(\sum _{k=1}^{n}a_{k}b_{k}\right)^{2} \leqslant \sum _{k=1}^{n}a_{k}^{2} \cdot \sum _{k=1}^{n}b_{k}^{2} .

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Problem 1294

Prove: the arithmetic mean of positive numbers is not greater than the mean of the squares of these numbers, that is,

\frac{1}{n}\sum _{k=1}^{n}x_{k} \leqslant \sqrt{\frac{1}{n}\sum _{k=1}^{n}x_{k}^{2}} .

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Problem 1295

Prove: the geometric mean of positive numbers is not greater than the arithmetic mean of these numbers, that is,

\sqrt[n]{x_{1}x_{2}\cdots x_{n}} \leqslant \frac{1}{n}(x_{1}+x_{2}+\cdots +x_{n}) .

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Problem 1296

(a)(b)(c)

Let $a$ and $b$ be two positive numbers; then the function defined by the equalities below

\text{if } s \neq 0, \Delta _{s}(a,b)=\left(\frac{a^{s}+b^{s}}{2}\right)^{\frac{1}{s}} ,

\Delta _{0}(a,b)=\lim _{s \to 0}\Delta _{s}(a,b)

is called the mean of order $s$ of the positive numbers $a$ and $b$ .

In particular, when $s=-1$ we get the harmonic mean, when $s=0$ we get the geometric mean (try to prove this!); when $s=1$ we get the arithmetic mean; when $s=2$ we get the quadratic mean.

Prove:

$\min (a,b) \leqslant \Delta_{s}(a,b) \leqslant \max (a,b)$ ;
when $a \neq b$ , the function $\Delta_{s}(a,b)$ is an increasing function of the variable $s$ ;
$\displaystyle \lim_{s \to -\infty }\Delta_{s}(a,b)=\min (a,b)$ ;
$\lim _{s \to +\infty }\Delta _{s}(a,b)=\max (a,b) .$

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(a)

(b)

(c)

Problem 1297

(a)(b)(c)

Let $f(x) \ (-\infty <x<+\infty )$ be a twice differentiable function, and

M_{k}=\sup _{-\infty <x<+\infty }\left|f^{(k)}(x)\right|<+\infty \ (k=0,1,2) .

Prove the inequality:

M_{1}^{2} \leqslant 2M_{0}M_{2} .

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(a)

(b)

(c)