Inverse Trigonometric Functions
Differentiation
Premium Question
Grade 12
Question:
Let $f(x) = x \cos^{-1}(\sin(-|x|))$ for $x \in (-\pi/2, \pi/2)$. Which is correct?
(1) $f'(0) = -\pi/2$
(2) $f'$ is decreasing on $(-\pi/2, 0)$, increasing on $(0, \pi/2)$
(3) $f$ is not differentiable at $x = 0$
(4) $f'$ is increasing on $(-\pi/2, 0)$, decreasing on $(0, \pi/2)$
Step-by-Step Solution
Key Concept: Use $\cos^{-1}(-t) = \pi - \cos^{-1} t$ to rewrite $f(x) = x(\pi/2 + |x|)$. Differentiate piecewise and check whether $f'$ is increasing or decreasing on each side.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (2)