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GATE 2024 EE – Question 43

Engineering Mathematics · Calculus: Maxima and minima · 2 marks · Multiple select

Let $f(t)$ be a real-valued function whose second derivative is positive for $-\infty<t<\infty$. Which of the following statements is/are always true?

  1. $f(t)$ has at least one local minimum.
  2. $f(t)$ cannot have two distinct local minima.
  3. $f(t)$ has at least one local maximum.
  4. The minimum value of $f(t)$ cannot be negative.

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Show answer and explanation

Correct answer: (B) $f(t)$ cannot have two distinct local minima.

Explanation

A strictly convex function has at most one local minimum: two distinct minima would need a local maximum between them, which would contradict $f''>0$. (B) is true. (A) is false ($e^t$ has none), (C) is false (a convex function has no local maximum), and (D) is false ($t^2-1$ has a negative minimum).