Question

The curve $f\left(x\right)=e^{x}sinx$ in the interval $[0,2\pi ]$. The value of $x$ for which the slope of the tangent drawn to the curve at $x$ is maximum, is

• A. $\frac{\pi }{4}$
• B. $\frac{\pi }{2}$
• C. $\frac{\pi }{6}$
• D. $\frac{\pi }{3}$

Answer 1

The correct option is B $\frac{\pi }{2}$

$g\left(x\right)=e^x,\sin x$

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

$f^{\prime }\left(x\right)$ is maximum

$\frac{d{f}^{\prime }\left(x\right)}{dx}=0$

$e^x(\cos x+\sin x)+e^x(\cos x-\sin x)=0$

$2x^x\cos x=0$

$\therefore \cos x=0\Rightarrow x=\frac{\pi }{2}$ or $\frac{3\pi }{2}$

$f^{\prime }\left(x\right)$ is maximum, $x=\frac{\pi }{2}$

Slope is maximum at $x=\frac{\pi }{2}$