Question

Number of points at which \( f(x) \) is not differentiable, where \[ f(x) = |\cos x| + 3 \text{in} [-\pi, \pi] \]

• A. 2
• B. 3
• C. 4
• D. None of these

Hint:

When dealing with absolute value functions, look for the points where the inside function equals zero, as these are the points where the function may not be differentiable due to corners.

Answer 1

The Correct Option is A

We are given the function: \[ f(x) = |\cos x| + 3 \] 

Step 1: Understand the behavior of \( f(x) \) The function \( f(x) \) involves \( |\cos x| \), which is the absolute value of \( \cos x \). The function \( \cos x \) is continuous and differentiable everywhere, except at points where \( \cos x = 0 \), because the absolute value function introduces a corner at those points, where the derivative is undefined. 

Step 2: Identify points where \( \cos x = 0 \) The function \( \cos x = 0 \) at \( x = \pm \frac{\pi}{2} \) within the interval \( [-\pi, \pi] \). At these points, \( |\cos x| \) will have a sharp corner, and the function \( f(x) = |\cos x| + 3 \) will not be differentiable. Thus, the points where \( f(x) \) is not differentiable are at \( x = \pm \frac{\pi}{2} \). 

Step 3: Conclusion The function is not differentiable at two points: \( x = \frac{\pi}{2} \) and \( x = -\frac{\pi}{2} \). 

Therefore, the number of points at which \( f(x) \) is not differentiable is 2. Thus, the correct answer is \( \boxed{2} \).