GATE DA 2024 Q37: Coefficients making a piecewise function differentiable

Let $f: \mathbb{R} \to \mathbb{R}$ be a function. Note: $\mathbb{R}$ denotes the set of real numbers.

$f(x) = \begin{cases} -x, & \text{if } x < -2 \\ ax^2 + bx + c, & \text{if } x \in [-2, 2] \\ x, & \text{if } x > 2 \end{cases}$

Which ONE of the following choices gives the values of $a, b, c$ that make the function $f$ continuous and differentiable?

Try it: multiple choice: exactly one correct option

Wrong answer costs 2/3 mark in GATE.

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Answer: A (official GATE 2024 final answer key)

Concept used: Continuity and differentiability at the joins: values and slopes must match from both sides.

First step: At $x = 2$ : $4a + 2b + c = 2$ and $4a + b = 1$ ; at $x = -2$ : $4a - 2b + c = 2$ and $-4a + b = -1$ .

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Source: official GATE 2024 DA question paper and final answer key, organised by IISc Bangalore. Question paper · Answer key