Let \(f: \mathbb{R} \to \mathbb{R}\) be a function such that \(f(x) = \max\{x, x^{3}\}\), \(x \in \mathbb{R}\), where \(\mathbb{R}\) is the set of all real numbers. The set of all points where \(f(x)\) is NOT differentiable is
Answer
(D)
Explanation
The curves \(y = x\) and \(y = x^{3}\) cross where \(x^{3} = x\), that is at \(x = -1, 0, 1\). Between these points the larger one switches:
- for \(x \lt -1\) and \(0 \lt x \lt 1\): \(f(x) = x\), slope 1;
- for \(-1 \lt x \lt 0\) and \(x \gt 1\): \(f(x) = x^{3}\), slope \(3x^{2}\).
At each crossing the slopes on the two sides differ: at \(x = \pm 1\) they are 1 and 3, and at \(x = 0\) they are 0 and 1. So the graph has a corner at each of these points, and \(f\) is not differentiable exactly on \(\{-1, 0, 1\}\). Elsewhere \(f\) equals a single polynomial near each point, so it is differentiable.
Answer: D