GATE CS 2024 Set 1 Q.11: Calculus

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ 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

$\{-1, 1, 2\}$

$\{-2, -1, 1\}$

$\{0, 1\}$

$\{-1, 0, 1\}$

Official answer and solution

Official answer: D
From: Official GATE 2024 answer key

Solution

Find the intersection points where $x = x^3$ and check whether the left-hand and right-hand derivatives match at each point.

Solving $x^3 = x$ gives $x(x - 1)(x + 1) = 0$ , so the two smooth functions $g(x) = x$ and $h(x) = x^3$ intersect at $x \in \{-1, 0, 1\}$ . Writing $f(x)$ piecewise:

f(x) = \begin{cases} x, & x < -1 \\ x^3, & -1 \le x < 0 \\ x, & 0 \le x < 1 \\ x^3, & x \ge 1 \end{cases}

Since $x$ and $x^3$ are differentiable everywhere on the open intervals, non-differentiability can only occur at the transition points $x \in \{-1, 0, 1\}$ . Comparing the derivatives $(x)' = 1$ and $(x^3)' = 3x^2$ :

At $x = -1$ : $f'_-(-1) = 1 \neq f'_+(-1) = 3(-1)^2 = 3$
At $x = 0$ : $f'_-(0) = 3(0)^2 = 0 \neq f'_+(0) = 1$
At $x = 1$ : $f'_-(1) = 1 \neq f'_+(1) = 3(1)^2 = 3$

Thus $f(x)$ is not differentiable at $\{-1, 0, 1\}$ .

Final answer

Option (D)

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2024 Computer Science and Information Technology - Set 1 (CS1)
Question: 11 of 65
Section: Computer Science and Information Technology

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