Question

The function $f(x) = |\cos x|$ is:

• A. Everywhere continuous and differentiable.
• B. Everywhere continuous but not differentiable at odd multiples of $\frac{\pi}{2}$.
• C. Neither continuous nor differentiable at $2n + 1, \, n \in \mathbb{Z}$.
• D. Not differentiable everywhere.

Answer 1

The Correct Option is B

The modulus function is continuous everywhere, and since $\cos x$ is continuous, $|\cos x|$ is also continuous everywhere.

However, $|\cos x|$ is not differentiable where $\cos x = 0$, which occurs at $x = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots$ (odd multiples of $\frac{\pi}{2}$).

Hence, the correct statement is that the function is everywhere continuous but not differentiable at odd multiples of $\frac{\pi}{2}$.