The Theorems of Rolle, Lagrange, and Cauchy. Increase and Decrease of Functions. Inequalities

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.

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

Prove that the Legendre polynomial

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

Prove that the Laguerre polynomial

has all its roots positive.

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

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.

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

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

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

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

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

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

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

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

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

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

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

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 $x$.

Prove the theorem: if

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

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

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

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

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

By direct squaring, prove the inequalities

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

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.

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.

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.

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

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

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

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

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

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

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

holds.

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

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

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

holds. Starting from it, derive the Cauchy inequality,

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

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

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