Continuity and Differentiability
Differentiability of Max/Min Functions
JEE Main 2020
Grade 12
Question:
Let $f : \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = \max\{x, x^2\}$. The set of all points where $f$ is not differentiable is:
(1) $\{-1, 1\}$
(2) $\{-1, 0\}$
(3) $\{0, 1\}$
(4) $\{-1, 0, 1\}$
Step-by-Step Solution
Key Concept: actually $\max(x, x^2)$: $x^2 > x$ when $x < 0$ or $x > 1$; $x > x^2$ when $0 < x < 1$. Corners at $x = 0$ (junction $x^2$ vs $x$: $f'(0^-) = 0$, $f'(0^+) = 1$) and $x = 1$ (junction $x$ vs $x^2$: $f'(1^-) = 1$, $f'(1^+) = 2$). Not differentiable at $\{0, 1\}$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (3)