Question

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

• A. $0$
• B. $\pi$
• C. $2\pi$
• D. $\frac{3\pi }{2}$

Answer 1

Answer: (B)

Let $m$ be the slope of the tangent to the curve $y=e^x\cos x$
Then, $m=\frac{dy}{dx}=e^x(\cos x-\sin x)$ diff w.r.t $x$
$\Rightarrow \frac{dm}{dx}=e^x(\cos x-\sin x)+e^x(-\cos x-\sin x)$ $=-2e^x\sin x$
and $\frac{d^{2}m}{d{x}^2}=-2e^x(\sin x+\cos x)$
Put $\frac{dm}{dx}=0$
$\Rightarrow \sin x=0$
$\Rightarrow x=0,\pi ,2\pi$
Clearly $\frac{d^{2}m}{d{x}^2}>0$ for $x=\pi$
Thus, $y$ is mininum at $x=\pi$
Hence, the value of $\alpha =\pi$.