GATE EE 2024 Set 1 — Question 43
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Engineering Mathematics → Calculus → Maxima & Minima
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Question
Let be a real-valued function whose second derivative is positive for . Which of the following statements is/are always true?
Correct answer
(B) f(t) cannot have two distinct local minima.
Solution
A function with for all is strictly convex.
1.Option (A): False. Consider . Here for all , but is strictly increasing and has no local minimum.
2.Option (B): True. For a strictly convex function, the derivative is strictly increasing. If has a root, it must be unique. Thus, there can be at most one local minimum, and it is impossible to have two distinct local minima.
3.Option (C): False. Strictly convex functions cannot have local maxima on an open interval. For example, has a minimum but no maximum.
4.Option (D): False. Consider . Here , but the minimum value is , which is negative.
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