The function f (x) = 2x ^{ 3 } β 3x ^{ 2 } β 12x + 4, has
(A) two points of local maximum
(B) two points of local minimum
(C) one maxima and one minima
(D) no maxima or minima
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Last updated at Dec. 4, 2021 by Teachoo
Question 25 The function f (x) = 2x3 β 3x2 β 12x + 4, has two points of local maximum (B) two points of local minimum (C) one maxima and one minima (D) no maxima or minima f (π₯) = 2π₯3 β 3π₯2 β 12π₯ + 4 Finding fβ (π) fβ (π) = 6π₯2 β 6π₯ β 12 = 6 (π₯"2 β" π₯" β 2" ) = 6 (π₯"2 β 2" π₯ "+ " π₯"β 2 " ) = 6 (π₯(π₯" β 2" )+1(π₯ "β 2" )) = 6 (π" + " π) (π "β" π) Putting fβ (π) = 0 6 (π₯+1) (π₯β2) = 0 β΄ π = β1, 2 For maxima or minima Finding fβ (π) fβ (π) = 12π₯ β 6 For π = β1 fβ (β1) = 12 (β1) β6 = β12 β 6 = β18 < 0 β΄ f has local maxima at x = β1 For π = 2 fβ (2) = 12 (2) β6 = 24 β 6 = 18 > 0 β΄ f has local minima at x = 2 Hence, π has one maxima and one minima. So, the correct answer is (C)