Question

Read the following statements.
Statement $1:$ Rolle's theorem holds for $f\left(x\right)=|\sin x|$ in $[0,2\pi ]$
Statement $2:$ Rolle's theorem does not hold for $g\left(x\right)=e^x\sin x$ in $[0,\pi ]$
which of the following option is correct.

• A. Both Statement $1$ ,Statement $2$ are true
• B. Both Statement $1$,Statement $2$ are false
• C. Statement $1$ is true, Statement $2$ is false
• D. Statement $1$ is false but Statement $2$ is true

Answer 1

The correct option is B Both Statement $1$,Statement $2$ are false

Statement $1:$ is false, because $f\left(x\right)=|\sin x|$ is not differentiable at $x=\pi$
Statement $2:g\left(x\right)=e^x\sin x,x\in [0,\pi ]$
$g\left(0\right)=0,g\left(\pi \right)=0$
$g\left(x\right)$ is continuous in $[0,\pi ]$ and differentiable in $(0,\pi )$
$\Rightarrow g^{\prime }\left(x\right)=e^x(\sin x+\cos x)=0$
$\Rightarrow x=\frac{3\pi }{4}\in (0,\pi )$
$\therefore$ Rolle's theorem holds for $g\left(x\right)$
Hence Statement $2$ is false.