GATE DA 2025 Set 1 — Question 51
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Question
Let be a twice-differentiable function and suppose its second derivative satisfies for all . Which of the following statements is/are correct?
Correct answer
(B) There does not exist x and y, x ≠ y, such that f'(x) = f'(y) = 0; (C) f has at most one global minimum; (D) f has at most one local minimum
Solution
Given for all , is strictly convex. This implies that is a strictly increasing function.
(A) Incorrect. Consider . , but has no local minima.
(B) Correct. Since is strictly increasing, it is injective. Thus . Therefore, there cannot be two distinct points with derivative 0.
(C) Correct. A strictly convex function can have at most one global minimum (if has a solution, it is the unique global minimum; otherwise there is none).
(D) Correct. Similarly, it can have at most one local minimum.
(A) Incorrect. Consider . , but has no local minima.
(B) Correct. Since is strictly increasing, it is injective. Thus . Therefore, there cannot be two distinct points with derivative 0.
(C) Correct. A strictly convex function can have at most one global minimum (if has a solution, it is the unique global minimum; otherwise there is none).
(D) Correct. Similarly, it can have at most one local minimum.
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