Consider the given function f(x). f(x) = cases ax + b & for x < 1 \\ x^3 + x^2 + 1 & for x ≥ 1 cases If the function is differentiable everywhere, the value of b must be ________. (rounded off to one decimal place)

GATE CSE 2025 Session 1 Q31: Value of b making a piecewise function differentiable

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)

Try it: numerical answer: type the value

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Numerical answer. No negative marking.

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Answer: -2.1 to -1.9 (official GATE 2025 final answer key)

Concept used: Differentiability at the join needs both continuity and equal one-sided derivatives.

First step: Match slopes at $x = 1$ : $a = 3x^2 + 2x$ at $x = 1$ ; then match values: $a + b = 1 + 1 + 1$ .

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Source: official GATE 2025 Session 1 CS question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key