R S Aggarwal Solutions for Exercise 3: EXERCISE

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 $x$.

Using Rolle’s theorem, find the point on the curve $y=x(x-4), x\in \left[0,4\right]$, where the tangent is parallel to the $x-\mathrm{axis}$.