Question

If Rolle’s theorem for $f\left(x\right)= e^{x} \left(sinx - cosx\right)$ is verified on $[\pi/4$, $5 \pi/4]$, then the value of $c$ is

• A. $\pi/3$
• B. $\pi/2$
• C. $3\pi/4$
• D. $\pi$

Answer 1

Answer: (D)

Given, $f (x) = e^{x} (sin \,x = cos\, x)$
on differentiating both sides w.r.t. , $x_{1}$ we get
$f'(x)=e^{x} \frac{d}{d x}(sin \,x-cos\, x)+(sin \,x-cos \,x) \frac{d}{dx}\left(e^{x}\right)$
[by using product rule of derivative]
$=e^{x}(cos\, x+sin \,x)+(sin\, x-cos \,x) e^{x}$
$=2 e^{x} \, sin \,x$
We know that, if Rolle's theorem is verified,
then their exist $c \in\left(\frac{\pi}{4}, \frac{5 \pi}{4}\right)$, such that $f' (c)=0$
$\therefore 2 e^{c} \,sin\, c=0$
$\Rightarrow sin\, c=0$
$\Rightarrow c=\frac{\pi}{2} \in\left(\frac{\pi}{4}, \frac{5 \pi}{4}\right)$