Question

The function $f(x)=(\sin x)^{\sin x}$ attains its minimum value when $x$ equal

• A. $\cos ^{-1}(0)$
• B. $\sin ^{-1}\left(\frac{1}{2}\right)$
• C. $\sin ^{-1}\left(\frac{2}{ e }\right)$
• D. $\sin ^{-1}\left(\frac{1}{ e }\right)$

Answer 1

Answer: (D)

$ y=(\sin x)^{\sin x}=e^{\sin x \ln (\sin x)}$
$\frac{d y}{d x}=(\sin x)^{\sin x}[\sin x \cdot \cot x+\ln (\sin x) \cdot \cos x]=0$
hence $\cos x[1+\ln (\sin x)]=0$
$\cos x=0$ or $\ln (\sin x)=-1$
$\sin x =\frac{1}{ e } ; x =\sin ^{-1} \frac{1}{ e } $ or $ x =2 n \pi+\frac{\pi}{2}$