Let f : R -> R be a twice-differentiable function and suppose its second derivative satisfies f''(x) > 0 for all x in R. Which of the following statements is/are ALWAYS correct? A. f has a local minima B. There does not exist x and y, such that f'(x) = 0 and f'(y) = 0 C. f has at most one global minimum D. f has at most one local minimum

GATE 2025 · Engineering Mathematics · Differentiation · medium

Answer: B, C, D

1. f' is strictly increasing: A strictly increasing function can take any value at most once. So f'(x) = 0 has at most one solution. This rules out having two distinct points x != y with f'(x) = 0 and f'(y) = 0.
2. Check option A: must f have a local minimum?: Option A is NOT always correct. f may have no local minimum at all.
3. Uniqueness of global and local minimum (Options C and D): Since f' can be zero at most once, f can have at most one critical point, hence at most one local minimum. If such a point exists it is the unique global minimum. So both C and D are ALWAYS correct.