Question

Is \[ f(x)=|x-2| \] differentiable at \(x=2\)?

• A. Yes
• B. No
• C. Only for \(x>2\)
• D. Only for \(x<2\)

Hint:

Functions of the form \(|x-a|\) are non-differentiable at \(x=a\).

Answer 1

The Correct Option is B

Concept: A function is differentiable at a point only if: \[ LHD=RHD \] Absolute value functions create a sharp corner where the expression inside modulus becomes zero.

Step 1: Find left hand derivative For: \[ x<2 \] \[ |x-2|=-(x-2) \] \[ =-x+2 \] Differentiate: \[ \frac{d}{dx}(-x+2)=-1 \] Thus: \[ LHD=-1 \]

Step 2: Find right hand derivative For: \[ x>2 \] \[ |x-2|=x-2 \] Differentiate: \[ \frac{d}{dx}(x-2)=1 \] Thus: \[ RHD=1 \]

Step 3: Compare derivatives \[ LHD=-1 \] \[ RHD=1 \] Since: \[ LHD\ne RHD \] the function is not differentiable at: \[ x=2 \] Final Answer: \[ \boxed{\text{No}} \]