Question

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

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

Answer 1

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

$\begin{array}{c}f\left(x\right)=e^x\sin x\\ \Rightarrow f^{\prime }\left(x\right)=e^x\cos x+\sin x{e}^x\\ \Rightarrow f^{\prime }\left(x\right)=e^x\left(\sin x+\cos x\right)\\ formaximumslope\\ \Rightarrow f^{\prime \prime }=e^x\left(\cos x-\sin x\right)+\left(\sin x+\cos x\right)e^x=0\\ \Rightarrow f^{\prime \prime }=2e^x\cos x=0\\ x=\pi /2\end{array}$