Question

If the slope of the tangent to the curve $y=e^x\cos x$ is minimum at $x=a,0\le a\le 2\pi$, then the value of $a$ is equal to

• A. $0$
• B. $\pi$
• C. $2\pi$
• D. $\frac{\pi }{4}$

Answer 1

The correct option is D $\frac{\pi }{4}$

For slope to be minimum,

$\frac{dy}{dx}=0$

$⟹\frac{dy}{dx}=e^x(\cos x-\sin x)$

$⟹e^a(\cos a-\sin a)=0$

$⟹\cos a=\sin a$

$⟹\tan \left(a\right)=1$

Hence, $a=\frac{\pi }{4}$