Discussion
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
\(f\) is not differentiable at \(\{-1, 0, 1\}\).
The curves \(y = x\) and \(y = x^{3}\) cross where \(x^{3} = x\), at x = −1, 0 and 1. Between the crossings the larger one switches:
- \(x \lt -1\): \(x \gt x^{3}\), so f = x
- \(-1 \lt x \lt 0\): f = \(x^{3}\)
- \(0 \lt x \lt 1\): f = x
- \(x \gt 1\): f = \(x^{3}\)
At each switch the two pieces have different slopes (1 against \(3x^{2}\), which is 3 at x = ±1 and 0 at x = 0), so each is a corner. The options containing 2 or −2 name points where f is smooth, and \(\{0, 1\}\) misses the corner at −1.
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