GATE | GATE CS 2008 | Question 25

A point on a curve is said to be an extremum if it is a local minimum or a local maximum. The number of distinct extrema for the curve 3x⁴ – 16x³ + 24x² + 37
(A) 0
(B) 1
(C) 2
(D) 3


Answer: (B)

Explanation: f(x) = 3x ⁴ – 16x³ + 24² +37

—> f ^‘(x) = 12x³ -48x² + 48x

—> f^”(x) = 36x² -96x + 48

f^‘(x)=0 —> =12x(x² – 4x +4) = 12x(x-2)²

f^‘(x) is negative for all x<0 and positive for all x>0

—> f(x) is decreasing to the left of 0 and increasing to the right of 0
—> f(x) has only one minimum (extrema) at x=0

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