I A Maron Solutions for Chapter: Application of Differential Calculus to Investigation of Functions, Exercise 13: Additional Problems

Does the function $f\left(x\right)=\left\{\begin{array}{lll}x& \mathrm{if} & x<1\\ \frac{1}{x}& \mathrm{if} & x\ge 1\end{array}\right.$ satisfy the conditions of the Lagrange theorem on the interval $\left[0,2\right]$ ?

Prove that for the function $y=\alpha x^2+\beta x+\gamma$ the number exists in the Lagrange formula, used on an arbitrary interval $\left[a,b\right]$,is the arithmetic mean of the numbers $a$ and $b:\xi =\frac{\left(a+b\right)}{2}$.

Prove that if the equation $a_0{x}^n+a_1{x}^{n-1}+\dots +a_{n-1}x=0$ has a positive root $x_0$, then the equation $n{a}_0{x}^{n-1}+\left(n-1\right)a_1{x}^{n-2}+\dots +a_{n-1}=0$ has a positive root less than $x_0$.

Prove that the equation $x^4-4x-1=0$ has two different real roots.

Prove that all roots of the derivative of the given polynomial $f\left(x\right)=\left(x+1\right)\left(x-1\right)\left(x-2\right)\left(x-3\right)$ are real.

Find a mistake in the following reasoning. The function 

$f\left(x\right)=\left\{\begin{array}{cc}{x}^2\sin \left(\frac{1}{x}\right)& \mathrm{for} x\ne 0\\ 0& \mathrm{for} x=0\end{array}\right.$ is differentiable for any $x$. By

Lagrange's theorem whence 

$\cos \frac{1}{\xi }=2\xi \sin \frac{1}{\xi }-x\sin \frac{1}{x} \left(0<\xi <x\right).$

As $x$ tends to zero $\xi$ will also tend to zero. Passing to the limit, we obtain $\underset{\xi \to 0}{\lim }\cos \left(\frac{1}{\xi }\right)=0$, whereas it is known that $\underset{x\to 0}{\lim }\cos \left(\frac{1}{x}\right)$ is non-existent.

Prove the following inequalities :

$\frac{a-b}{a}<\ln \frac{a}{b}<\frac{a-b}{b}$ if $0<b<a$

Prove the following inequalities :

$p{y}^{p-1}\left(x-y\right)<x^p-y^p<p{x}^{p-1}\left(x-y\right)$ if $0<y<x$ and $p>1$.