The GATE Grind

GATE 2025 DA – Question 51

Calculus and Optimization · Taylor series, maxima and minima, single-variable optimization · 2 marks · Multiple select

Let $f: \mathbb{R} \to \mathbb{R}$ be a twice-differentiable function and suppose its second derivative satisfies $f''(x) > 0$ for all $x \in \mathbb{R}$. Which of the following statements is/are ALWAYS correct?

  1. $f$ has a local minima
  2. There does not exist $x$ and $y$, $x \ne y$, such that $f'(x) = f'(y) = 0$
  3. $f$ has at most one global minimum
  4. $f$ has at most one local minimum

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Show answer and explanation

Correct answer: (B) There does not exist $x$ and $y$, $x \ne y$, such that $f'(x) = f'(y) = 0$; (C) $f$ has at most one global minimum; (D) $f$ has at most one local minimum

Explanation

Since $f'' > 0$, $f'$ is strictly increasing, so it can be zero at no more than one point (B is true), and a strictly convex function has at most one point where it reaches a minimum, and any local minimum is that one (C and D are true). A is not guaranteed: $f(x) = e^x$ has $f'' > 0$ everywhere but no minimum at all.