GATE CE 2022 Set 2 — Question 36
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Engineering Mathematics → Calculus → Local Maxima & Minima
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Question
Consider the polynomial on the domain given by . The first and second derivatives are and .Consider the following statements:
I. The given polynomial is zero at the boundary points and .
II. There exists one local maxima of within the domain .
III. The second derivative throughout the domain .
IV. There exists one local minima of within the domain .The correct option is:
I. The given polynomial is zero at the boundary points and .
II. There exists one local maxima of within the domain .
III. The second derivative throughout the domain .
IV. There exists one local minima of within the domain .The correct option is:
Correct answer
(B) Only statements I, II and IV are correct.
Solution
Given for .Statement I:
Statement I is True.Statements II and IV:
Setting to find critical points:
Both and lie within the domain .
At , Local Maxima.
At , Local Minima.
Thus, there is one local maxima and one local minima within the domain. Statements II and IV are True.Statement III:
. At , , which is not . Thus, is not positive throughout the domain. Statement III is False.Since statements I, II, and IV are correct, the correct option is (B).
Statement I is True.Statements II and IV:
Setting to find critical points:
Both and lie within the domain .
At , Local Maxima.
At , Local Minima.
Thus, there is one local maxima and one local minima within the domain. Statements II and IV are True.Statement III:
. At , , which is not . Thus, is not positive throughout the domain. Statement III is False.Since statements I, II, and IV are correct, the correct option is (B).
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