Discussion
The function \(f(x) = |x|^{3}, x \in \mathbf{R}\) is :
Answer: C
\(f(x) = |x|^{3}\) is of class \(c^{2}\) but not \(c^{3}\).
Write \(f(x) = x^{3}\) for \(x \ge 0\) and \(-x^{3}\) for \(x \lt 0\). Then
\[ f'(x) = 3x|x|, \qquad f''(x) = 6|x| \]
Both are continuous everywhere, including at 0 (where each equals 0), so f is twice continuously differentiable. But \(f''(x) = 6|x|\) has a corner at 0: its one-sided derivatives are \(+6\) and \(-6\), so \(f'''(0)\) does not exist and f is not of class \(c^{3}\).
'Continuous but not differentiable' and 'differentiable but not \(c^{1}\)' stop too early: the cube flattens the corner of \(|x|\) twice over.
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