Question

Which of the following is differentiable at \(x=0\)?

• A. \(f(x)=\cos|x|+|x|\)
• B. \(f(x)=\sin|x|+|x|\)
• C. \(f(x)=\cos|x|-|x|\)
• D. \(f(x)=\sin|x|-|x|\)

Hint:

For functions involving \(|x|\), always replace \(|x|\) by \(x\) for \(x\gt 0\) and by \(-x\) for \(x\lt 0\), then compare the left and right derivatives.

Answer 1

The Correct Option is D

Step 1: Check LHD and RHD at x=0 for each option.
Note \(|x|\) has LHD \(=-1\) and RHD \(=+1\) at 0, while \(\sin|x|\) has LHD \(=-1\) and RHD \(=+1\), and \(\cos|x|\) has derivative \(0\) from both sides at \(0\).

Step 2: Test Option 4: f(x)=sin|x|-|x|.
RHD: \(\sin'(0^+)-1=1-1=0\). LHD: \(-\sin'(0^+)-(-1)=-1+1=0\). Both derivatives equal \(0\), so \[\boxed{f(x)=\sin|x|-|x|\text{ is differentiable at }x=0}\]