Question

The point in the interval $[0,2\pi ]$ where $f\left(x\right)=e^{x}sinx$ has maximum slope, is

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

Answer 1

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

We have, $f^{\prime }\left(x\right)=e^x+cosx+sinx{e}^{x}And{f}^{\prime }\left(x\right)=-sinx{e}^x+cosx{e}^x+cosx{e}^x+sinxcosx{e}^x.Now,f^{\prime }\left(x\right)=2cosxcosx{e}^x=0\Rightarrow cosx=0\Rightarrow x=\frac{\pi }{2}.Also,f^{\prime }\left(x\right)=-2sinx{e}^x+2cosx{e}^x=-ve$
$\therefore$ Slope is maximum at $x=\frac{\pi }{2}.$
Hence (b) is the correct answer.