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

There does not exist x and y, x y, such that f'(x) = f'(y) = 0

f has at most one global minimum

f has at most one local minimum

There does not exist x and y, x y, such that f'(x) = f'(y) = 0

f has at most one global minimum

f has at most one local minimum

GATE DA 2025 Q51: Consequences of f″(x) > 0 everywhere

Calculus and Optimization
2 marks
MSQ
No negative marking

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ 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?

Try it: multiple select: one or more correct options

Select all correct options. No negative marking, no partial marks.

Show answer and explanation

Answer: B, C, D (official GATE 2025 final answer key)

Concept used: Strict convexity: $f'$ is strictly increasing.

First step: Since $f'' > 0$ , $f'$ is strictly increasing, so it can be zero at most once. For option A, look for a strictly convex function with no minimum, such as $e^x$ .

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

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GATE DA 2025 Q51: Consequences of f″(x) > 0 everywhere

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