GATE CS 2023 Set 1 — Question 28
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Engineering Mathematics → Calculus → Maxima & Minima
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Question
Letbe a real-valued function.Which of the following statements is/are TRUE?
Correct answer
(B) f(x) has a local maximum.; (D) f(x) has a local minimum.
Solution
To find the local extrema of the function , we first find its first derivative:
.Setting gives critical points at and .Next, we find the second derivative to test for local maxima and minima:
.Evaluating the second derivative at the critical points:
.Setting gives critical points at and .Next, we find the second derivative to test for local maxima and minima:
.Evaluating the second derivative at the critical points:
1.At : . Since the second derivative is negative, has a local maximum at .
2.At : . Since the second derivative is positive, has a local minimum at .
Therefore, has both a local maximum and a local minimum. Statements (B) and (D) are TRUE.Continue learning with Success Tracker
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