Question

The slope of the curve $y=e^x\cos x,x\in (-\pi ,\pi )$ is maximum at

• A. $x=\frac{\pi }{2}$
• B. $x=-\frac{\pi }{2}$
• C. $x=\frac{\pi }{4}$
• D. $x=0$
• E. $x=\frac{\pi }{3}$

Answer 1

The correct option is D $x=0$

Let $m$ be the slope of the tangent to the curve
$y=e^x\cos x$
Then, $m=\frac{dy}{dx}=e^x(\cos x-\sin x)$
$\Rightarrow \frac{dm}{dx}=e^x(\cos x-\sin x)+e^x(-\sin x-\cos x)=-2e^x\sin x$
and
$\frac{d^{2}m}{d{m}^2}=-2e^x\left(\sin x+\cos x\right)$
$\therefore \frac{dm}{dx}=0$
$\Rightarrow \sin x=0\Rightarrow x=0$
clearly, $\frac{d^{2}m}{d{m}^2}<0$