Question

Consider the function $f(x)=\sin ^5 x+\cos ^5 x-1, x \in\left[0, \frac{\pi}{2}\right]$. Which of the following is/are correct?

• A. $f$ is monotonic increasing in $\left(0, \frac{\pi}{4}\right)$.
• B. $f$ is monotonic decreasing in $\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$.
• C. $\exists$ some $c \in\left(0, \frac{\pi}{2}\right)$ for which $f ^{\prime}( c )=0$
• D. The equation $f(x)=0$ has two roots in $\left[0, \frac{\pi}{2}\right]$.

Answer 1

Answer: (C), (D)

We have $f^{\prime}(x)=5 \sin ^4 x \cos x-5 \cos ^4 x \sin x=5 \sin x \cos x(\sin x-\cos x)(1+\sin x \cos x)$
$\therefore f ^{\prime}( x )=0$ at $x =\frac{\pi}{4} . $ Also $f ^{\prime}(0)= f ^{\prime}\left(\frac{\pi}{2}\right)=0 $
Hence $\exists$ some $c \in\left(0, \frac{\pi}{2}\right)$ for which $f ^{\prime}( c )=0$ (By Rolle's Theorem) $ \Rightarrow $ (C) is correct.
Also in $\left(0, \frac{\pi}{4}\right) f$ is decreasing and in $\left(\frac{\pi}{4}, \frac{\pi}{2}\right) f$ is increasing $\Rightarrow$ minimum at $x =\frac{\pi}{4}$
As $ f(0)=f\left(\frac{\pi}{2}\right)=0 \Rightarrow 2$ roots $\Rightarrow $ (D) is correct.