Let f(x) be a polynomial of degree 5 such that x= 1 are its critical points. If _ x 0 (2+ f(x) x^3 )=4, then which one of the following is not ture?

JEE Main 2020 Mathematics question

Mathematics Application of Derivatives JEE Main 2020

Let f(x) be a polynomial of degree 5 such that are its critical points. If , then which one of the following is not ture?

(A)f is an odd function
(B)
(C)x = 1 is a point of minima and x = –1 Is a point of maxima of fCorrect
(D)x = 1 is a point of maxima and x = –1 Is a point of minimum of f

Answer: (C)

Solution

f'(x) = a x²(x² − 1) and the limit gives f(x) = −(6/5)x⁵ + 2x³. Then f is odd, f(1) − 4f(−1) = 4/5 + 16/5 = 4, and f' > 0 on (−1, 1), f' < 0 outside, so

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Question ID VK-010880