Verify Rolle's theorem for the following function: $f\left(x\right)={\mathrm{e}}^{-x}(\sin x-\cos x)$ in $\left[\frac{\pi }{4},\frac{5\mathrm{\pi }}{4}\right]$

Verify Rolle's theorem for the following function: $f\left(x\right)=\sin x-\sin 2x$ in $\left[0,2\pi \right]$.

Verify Rolle's theorem for the following function: $f\left(x\right)=x(x+2)e^x$ in $\left[-2,0\right]$.

Show that $f\left(x\right)=x(x-5{)}^2$ satisfies Rolle’s theorem on $[0, 5]$ and that the value of $c$ is $\frac{5}{3}$.

Discuss the applicability for Rolle’s theorem, when: $f\left(x\right)=(x-1)(2x-3)$, where $1\le x\le 3$ 

Discuss the applicability for Rolle’s theorem, when: $f\left(x\right)=x^{\frac{1}{2}}$ on $\left[-1,1\right]$

Discuss the applicability for Rolle’s theorem, when: $f\left(x\right)=2+(x-1{)}^{\frac{2}{3}}$ on $\left[0,2\right]$.

Discuss the applicability for Rolle’s theorem, when:

$f\left(x\right)=\cos \frac{1}{x}$ on $\left[-1,1\right]$

Discuss the applicability for Rolle’s theorem, when:

$f\left(x\right)=\left[x\right]$ on $\left[-1,1\right]$, where $\left[x\right]$ denotes the greatest integer not exceeding $x$.