Parthasarathi Mukhopadhyay and Manabendra Nath Mukherjee Solutions for Exercise 1: EXERCISE

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

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

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

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

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 $f\left(x\right)=x\left(x+2\right)e^{-\frac{x}{2}}$ in $\left[-2, 0\right]$.

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