Question:

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

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Differentiability and Absolute Values}
If \( f(x) \) involves \( |x| \), check for corner points at \( x = 0 \)
Constant functions are always differentiable
Visualize or test left-hand and right-hand derivatives for piecewise cases
Updated On: May 19, 2025
  • \( f(x) = \cos(|x| + |x|) \)
  • \( f(x) = \sin(|x| + |x|) \)
  • \( f(x) = \cos(|x| - |x|) \)
  • \( f(x) = \sin(|x| - |x|) \)
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The Correct Option is D

Solution and Explanation

\[ f(x) = \sin(|x| - |x|) = \sin(0) = 0 \Rightarrow f(x) = 0 \text{ for all } x \] \[ \Rightarrow f(x) \text{ is a constant function } \Rightarrow \text{Differentiable everywhere} \] Other functions involve absolute values inside trigonometric functions and result in a non-smooth corner at \( x = 0 \), which causes non-differentiability.
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