Consider the given function f(x). f(x) = cases ax + b & for x < 1 \\ x3 + x2 + 1 & for x 1 cases If the function is differentiable everywhere, the value of b must be … (rounded off to one decimal place)

Official answer: -2. Explanation: For the piecewise function f(x) to be differentiable everywhere, it must be differentiable and continuous at the boundary point x = 1. 1. Differentiability at x = 1: The left-hand derivative (LHD) must equal the right-hand derivative (RHD) at x = 1: f'(1-) = . ddx(ax + b) |x=1 = a f'(1+) = . ddx(x3 + x2 + 1) |x=1 = . (3x2 + 2x) |x=1 = 3(1)2 + 2(1) = 5 Equating the derivatives gives: a = 5 2. Continuity at x = 1: Since differentiability implies continuity, the left-hand limit must equal the value of the function at x = 1: x 1- f(x) = f(1) a(1) + b = (1)3 + (1)2 + 1 a + b = 3 Substitute a = 5 into the continuity equation: 5 + b = 3 b = -2 Thus, the value of b is -2.

Engineering Mathematics · Calculus

Official IIT answer key · IIT Guwahati · Audited Aug 2026

Take topic test

GATE CSE 2025 Set 1 Q31 · NAT · 1 mark

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)

Numerical answer

Enter a value to check

Key concept

Calculus

Step-by-step verified solution

Examiner trap & misconception

Topic notes

How GATE tests Calculus

Calculus in GATE CS tests single-variable differential and integral calculus, foundational limit evaluations, continuity and differentiability of piecewise/absolute-value functions, and mean value theorems (Rolle's and LMVT). Earlier years (2012-2015) also included numerical methods such as Newton-Raphson, Bisection, and Trapezoidal/Simpson's rules, whereas recent papers (2020-2026) focus heavily on analytical properties: odd/even symmetry in single and multiple integrals, boundary extrema, points of non-differentiability, and MSQ-based structural analyses of functions.

Full Calculus guide →

More on Calculus

All 40 questions on Calculus →

Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.

Engineering Mathematics · Calculus

Official IIT answer key · IIT Guwahati · Audited Aug 2026

Take topic test

GATE CSE 2025 Set 1 Q31 · NAT · 1 mark

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)

Numerical answer

Enter a value to check

Key concept

Calculus

Step-by-step verified solution

Examiner trap & misconception

Topic notes

How GATE tests Calculus

Full Calculus guide →

More on Calculus

All 40 questions on Calculus →

Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.

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)

Step-by-step verified solution

Check you can still solve Calculus under a clock

More from GATE CSE 2025 Set 1

More on 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)

Step-by-step verified solution

Check you can still solve Calculus under a clock

More from GATE CSE 2025 Set 1

More on Calculus

Step-by-step verified solution

Check you can still solve Calculus under a clock

More from GATE CSE 2025 Set 1

More on Calculus

Share this solution

Step-by-step verified solution

Check you can still solve Calculus under a clock

More from GATE CSE 2025 Set 1

More on Calculus

Share this solution