Continuity and Differentiability
Differentiability
JEE Advanced 2015 Paper 1
Grade 12
Question:
Let $g : \mathbb{R} \to \mathbb{R}$ be differentiable with $g(0) = 0$, $g'(0) = 0$ and $g'(1) \neq 0$. Define
$$f(x) = \begin{cases} \frac{x}{|x|} g(x), & x \neq 0 \\ 0, & x = 0 \end{cases}$$
Then $f$ is differentiable at $x = 0$ if:
(1) $g$ is differentiable and $g'(0) = 0$
(2) $g'(1) \neq 0$
(3) $g(0) = 0$ only
(4) $f$ is never differentiable at $x = 0$
Step-by-Step Solution
Key Concept: $f'(0) = \lim_{h \to 0} f(h)/h = \lim_{h \to 0} g(h)/|h|$. Since $g(0) = g'(0) = 0$, $g(h) = O(h^2)$, so $g(h)/|h| = O(|h|) \to 0$. Thus $f$ is always differentiable at $x = 0$ given the stated conditions. Answer: (1) captures the key condition.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)