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 :

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

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

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

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

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

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