GATE AG 2012 • Q32 MCQ • 1 Mark

Section 1: Engineering Mathematics

Calculus

Subtopic: Maxima and minima of function with several independent variables

Question Statement:

If f'(x) = eˣ and f(0) = 5, then from the Mean Value Theorem, the value of f(1) lies between:

Options:

  • (A) 2 and (2 + e)
  • (B) 3 and (2 + e)
  • (C) 3 and (3 + e)
  • (D) 6 and (5 + e)

Official Answer Key:

D

Step-by-Step Solution & Explanation

By MVT, f(1) - f(0) = f'(c) for some c ∈ (0,1), where f'(c) = e^c lies strictly between f'(0)=1 and f'(1)=e (since e^x is increasing). So f(1) - 5 lies strictly between 1 and e, i.e., f(1) lies between 6 and (5 + e). Answer: D.

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