GATE EC 2025 Set 1 — Question 25
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Engineering Mathematics → Calculus → Maxima & Minima
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Question
Consider the function , defined as . Which of the following statements is/are correct? (Here, is the set of real numbers.)
Correct answer
(A) f has no global maximizer; (B) f has no global minimizer
Solution
To determine the properties of the function , we first find its first and second derivatives.
To find critical points, set :
So, the critical points are and .
For a cubic polynomial with a positive leading coefficient, as , , and as , . Therefore, the function does not have a global maximizer or a global minimizer.
1.First Derivative ():
To find critical points, set :
So, the critical points are and .
2.Second Derivative ():
3.Local Extrema using the Second Derivative Test:
- At : . Since , is a local maximizer.
- At : . Since , is a local minimizer.
For a cubic polynomial with a positive leading coefficient, as , , and as , . Therefore, the function does not have a global maximizer or a global minimizer.
5.Evaluating the Options:
- (A) has no global maximizer: This is TRUE.
- (B) has no global minimizer: This is TRUE.
- (C) is a local minimizer of : This is FALSE; is a local maximizer.
- (D) is a local maximizer of : This is FALSE; is a local minimizer.
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