Let $f(x) = sin^{-1}(\cos x)$ for $x \in \mathbb{R}$. Which of the following statements is/are CORRE

Question

Let $f(x) = sin^{-1}(\cos x)$ for $x \in \mathbb{R}$. Which of the following statements is/are CORRECT?

Professor PC Thomas Classes · Answer

Applying the chain rule, $f'(x) = \frac{1}{\sqrt{1 - \cos^2 x}} \cdot (-\sin x) = \frac{-\sin x}{|\sin x|}$. This implies $f'(x) = -1$ when $\sin x > 0$ (e.g., $x \in (0, \pi)$) and $f'(x) = 1$ when $\sin x < 0$ (e.g., $x \in (-\pi, 0)$). Thus, (B) is correct. At $x = n\pi$, $\sin x = 0$, the left-hand and right-hand derivatives alternate between $1$ and $-1$, making the function continuous but non-differentiable at these points. This makes (A) correct. Evaluating the left-hand derivative at $x = 0$ implies analyzing $x 	o 0^-$, where $\sin x < 0$, yielding $f'(x) = 1$. Thus, (C) is correct. For (D), checking the period: $f(x+\pi) = \arcsin(\cos(x+\pi)) = \arcsin(-\cos x) = -\arcsin(\cos x) = -f(x)$. The function is not periodic with period $\pi$, its fundamental period is $2\pi$. Thus (D) is incorrect.