Question

If we apply the Rolle's theorem to $f\left(x\right)=e^x\sin x$, $x\in [0,\pi ]$ then $c=0$

• A. $3\pi /4$
• B. $5\pi /4$
• C. $\pi /4$
• D. $7\pi /4$

Answer 1

The correct option is A $3\pi /4$

We know by Rolle's theorem that a function $f\left(x\right)$ is continuous on $[a,b]$ and differentiable in $(a,b)$ then there exists $C\in (a,b)$ such that $f^{\prime }\left(c\right)=0$

Here $f\left(x\right)=e^x.\sin xx\in [0,\pi ]$

$f^{\prime }\left(x\right)=e^x\sin x+e^x\cos x$

$f^{\prime }\left(c\right)=e^c(\sin c+\cos c)=0$

Since $e^c\ne 0$, we have

$\tan c=-1\Rightarrow c=\frac{3\pi }{4}$