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Question: Answered & Verified by Expert
If $\mathrm{f}(\mathrm{x})$ is differentiable everywhere,then which one of the following is correct?
MathematicsContinuity and DifferentiabilityNDANDA 2007 (Phase 1)
Options:
  • A $|\mathrm{f}|$ is differentiable everywhere
  • B $|\mathrm{f}|^{2}$ is differentiable everywhere
  • C $\mathrm{f}|\mathrm{f}|$ is not differentiable at some points
  • D $\text { None of the above }$
Solution:
2752 Upvotes Verified Answer
The correct answer is: $\mathrm{f}|\mathrm{f}|$ is not differentiable at some points
If $\mathrm{f}(\mathrm{x})$ is differential everywhere then $\mid \mathrm{f}$ | is not differentiable at some point, so, $\mathrm{f}|\mathrm{f}|$ is not differentiable at some point. [Example: $\mathrm{f}(\mathrm{x})=\mathrm{x}$ is differentiable everywhere but $|\mathrm{f}(\mathrm{x})|=|\mathrm{x}|$ is not differentiable at $\mathrm{x}=0]$

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