JEE PYQ: Application Of Derivatives Question 48
Question 48 - 2020 (08 Jan Shift 1)
Let $f(x) = x\cos^{-1}(-\sin|x|)$, $x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, then which of the following is true?
(1) $f’$ is increasing in $\left(-\frac{\pi}{2}, 0\right)$ and decreasing in $\left(0, \frac{\pi}{2}\right)$
(2) $f’(0) = -\frac{\pi}{2}$
(3) $f$ is not differentiable at $x = 0$
(4) $f’$ is decreasing in $\left(-\frac{\pi}{2}, 0\right)$ and increasing in $\left(0, \frac{\pi}{2}\right)$
Show Answer
Answer: (4)
Solution
$f’(x) = x(\frac{\pi}{2} + |x|)$. For $x \ge 0$: $f(x) = x(\frac{\pi}{2} + x)$, $f’(x) = \frac{\pi}{2} + 2x$. For $x < 0$: $f(x) = x(\frac{\pi}{2} - x)$, $f’(x) = \frac{\pi}{2} - 2x$. $f’$ is decreasing on $(-\frac{\pi}{2}, 0)$ and increasing on $(0, \frac{\pi}{2})$.
BETA
