Question

The function \(f(x)=|x-2|+|x|+|x+2|\) is not differentiable at which points?

• A. \(x=-2\)
• B. \(x=0\)
• C. \(x=2\)
• D. \(x=-2,0,2\)

Hint:

A modulus function \(|x-a|\) is not differentiable at \(x=a\). For a sum of modulus functions, check all such turning points.

Answer 1

The Correct Option is D

Concept:
The modulus function \(|x-a|\) is not differentiable at \(x=a\), because its graph has a sharp corner at that point.

Step 1: Write the given function. \[ f(x)=|x-2|+|x|+|x+2| \]

Step 2: Identify the critical points.
Each modulus term becomes zero at a particular point. For: \[ |x-2| \] critical point is: \[ x=2 \] For: \[ |x| \] critical point is: \[ x=0 \] For: \[ |x+2| \] critical point is: \[ x=-2 \]

Step 3: Differentiability conclusion.
The function may fail to be differentiable at points where any modulus expression changes sign. Thus, \(f(x)\) is not differentiable at: \[ x=-2,0,2 \] \[ \therefore \text{Correct Answer is (D)} \]