GATE CS 2025 Set 1 Q.31: Calculus

Consider the given function $f(x)$ .

f(x) = \begin{cases} ax + b & \text{for } x < 1 \\ x^3 + x^2 + 1 & \text{for } x \ge 1 \end{cases}

If the function is differentiable everywhere, the value of $b$ must be ________. (rounded off to one decimal place)

Your answer

Official answer and solution

Solution

Enforce both continuity and equality of left-hand and right-hand derivatives at $x = 1$ .

For $f(x)$ to be differentiable at $x = 1$ , it must be both continuous and have equal left-hand and right-hand derivatives at $x = 1$ :

Equal derivatives at $x = 1$ :
- Left-hand derivative: $f'_-(1) = a$
- Right-hand derivative: $f'_+(1) = \left.(3x^2 + 2x)\right|_{x=1} = 3(1)^2 + 2(1) = 5$
- Thus $a = 5$ .
Continuity at $x = 1$ :
- $\lim_{x \to 1^-} (ax + b) = f(1) \implies a(1) + b = 1^3 + 1^2 + 1 = 3$
- Substituting $a = 5$ gives $5 + b = 3 \implies b = -2$ .

Final answer

-2

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

Paper: GATE 2025 Computer Science and Information Technology - Session 1 (CS1)
Question: 31 of 65
Section: Computer Science and Information Technology

GATE questions are the work of the organising institutes. This site is independent and is not affiliated with them.