GATE AG 2020 • Q11 MCQ • 1 Mark

Section 1: Engineering Mathematics

Calculus

Subtopic: Maxima and minima of function with several independent variables

Question Statement:

The function $f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1$ has a:

Options:

  • (A) Maxima at x = 0
  • (B) Minima at x = 0
  • (C) Maxima at x = 1
  • (D) Minima at x = 1

Official Answer Key:

D

Step-by-Step Solution & Explanation

f(x) is exactly the binomial expansion of (x−1)⁴. f′(x)=4(x−1)³=0 ⇒ x=1 is the only critical point; since (x−1)⁴ ≥ 0 always (with equality only at x=1), this is a minimum (value 0) at x=1.

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