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

1. Understand the problem:

We need to analyze the continuity and differentiability of the function \( f(x) = |\cos x| \).

2. Analyze continuity:

The cosine function is continuous everywhere, and the absolute value function is also continuous. Therefore, \( f(x) = |\cos x| \) is continuous everywhere.

3. Analyze differentiability:

The absolute value function is not differentiable where its argument is zero. For \( f(x) = |\cos x| \), this occurs at:

\[ x = (2n + 1)\frac{\pi}{2}, \quad n \in \mathbb{Z} \]

At these points, the left and right derivatives are not equal, making the function non-differentiable.

Correct Answer: (B) Everywhere continuous but not differentiable at odd multiples of \(\frac{\pi}{2}\)

Answer 2

1. Continuity:

• The cosine function ($ \cos(x) $) is continuous everywhere.
• The absolute value function ($ |x| $) is continuous everywhere.
• The composition of two continuous functions is continuous. Therefore, $ |\cos(x)| $ is continuous everywhere.

2. Differentiability:

• $ \cos(x) $ is differentiable everywhere.
• However, the absolute value function introduces points where differentiability may fail. These are the points where the function inside the absolute value is zero (because the absolute value function has a sharp corner at $ x = 0 $).

3. Zeros of $ \cos(x) $:

$ \cos(x) = 0 $ at $ x = (2n+1)\frac{\pi}{2} $, where $ n $ is an integer.

4. Consider $ |\cos(x)| $:

• Take the derivative around $ x = \frac{\pi}{2} $.

Therefore, the function $ f(x) = |\cos x| $ is everywhere continuous but not differentiable at odd multiples of $ \frac{\pi}{2} $.