Rolle's Theorem

The function for which the Rolle's theorem is applicable is:

The value of ${}^{\text{'}}c^{\text{'}}$  for which $f^{\text{'}}\left(c\right)=0$ in case Rolle's theorem is applied to $f\left(x\right)=\cos x$ in $\left[-\frac{\pi }{2}, \frac{\pi }{2}\right]$ is

The Rolle's theorem is _____ (applicable / not applicable) to $f\left(x\right)=\left[x\right]$ in $\left[-3, 3\right]$.

The Rolle's theorem is _____ (applicable / not applicable) to $f\left(x\right)=\sqrt{x}$ in $\left[-1, 1\right]$.

Rolle's theorem is _____ (applicable / not applicable) to $f\left(x\right)=\cos \frac{1}{x}$ in $\left[-1, 1\right]$.

Rolle's theorem is applicable to $f\left(x\right)=x\left(x+2\right)e^{-\frac{x}{2}}$ in $\left[-2, 0\right]$.

Rolle's theorem is applicable to $f\left(x\right)=x^{\frac{2}{3}}$ in $\left[-1, 1\right]$.

Rolle's theorem is applicable to $f\left(x\right)=\tan x$ in $\left[0, \pi \right]$.

If at least one condition of Rolle's theorem is not satisfied by a function $f$ on $\left[a, b\right]$, then $f^{\text{'}}\left(c\right)\ne 0$ for any $c\in \left(a, b\right)$.

Conclusion of the Rolle's theorem may be true for a function $f$ in $\left[a, b\right]$, though all the conditions of the theorem are not satisfied by $f$ on $\left[a, b\right]$.

Rolle's theorem is always applicable to any continuous function $y=f\left(x\right)$ in $\left[a, b\right]$ for which $f\left(a\right)=f\left(b\right)$ holds.

The value of $c\in \left(\frac{1}{2}, 3\right)$ is $x=$_____ when Rolle's theorem is applied on $f\left(x\right)=2x^3-5x^2-4x+3$ in $\left[\frac{1}{2}, 3\right]$.

The conclusion of Rolle's theorem holds at the point $x=$_____, when it is applied on $x^2-x-12$ in $\left[-3, 4\right]$.

Rolle's theorem is _____ (applicable / not applicable) to $f\left(x\right)=\left|\cot x\right|$  in $\left[-\frac{\pi }{2}, \frac{\pi }{2}\right]$.

Considering the function $f\left(x\right)=\left(x-4\right)\log x$ in $\left[1, 4\right]$ show that $x \log x=4-x$ is satisfied by at least one value of $x$ in between $1$ and $4$

If $f\left(x\right)$ is a polynomial in $x$, then prove that between any two roots of $f\left(x\right)$, there exists a root of $f^{\text{'}}\left(x\right)$.

For the function $f\left(x\right)=\left|x\right|+\left|x-1\right|$ in $\left[-1, 2\right]$, verify that the conditions of Rolle's theorem are not satisfied but the conclusion is true.

If Rolle's theorem is applied to the function $f\left(x\right)=3x^3-11x^2-6x+8$  in $\left[\frac{2}{3}, 4\right]$, find the value of $c$ where

Examine the validity of the hypothesis and the conclusion of Rolle's theorem for the following functions:
$f\left(x\right)=\frac{1}{x}+\frac{1}{1-x}$ in $\left[0, 1\right]$

Examine the validity of the hypothesis and the conclusion of Rolle's theorem for the following function:
$f\left(x\right)=\frac{1}{2-x^2}$ in $\left[-1, 1\right]$