question_answer

The value of x in the interval \[[0,\,\,2\pi ),\] where  \[f(x)={{e}^{x}}.\,\,sin\,x\] has maximum slope is -

A) \[\frac{\pi }{4}\]

B) \[\frac{\pi }{2}\]

C) \[\pi \]

D) \[\frac{3\pi }{2}\]

Correct Answer: B

Solution :

\[\because f(x)={{e}^{x}}.\sin x\]

\[f'(x)={{e}^{x}}\sin x+{{e}^{x}}\sin x\]

\[f''(x)={{e}^{x}}\sin x+2{{e}^{x}}\cos x-{{e}^{x}}\sin x\]

\[=2{{e}^{x}}\cos x=0\]

\[\Rightarrow x=\frac{\pi }{2},\frac{3\pi }{2}\]

\[f'(x)\]is maximum at \[x=\pi /2\]