If $ x=1 $ is a critical point of the function $ f(x)=\left(3 x^{2}+a x-2-a\right) e^{x} $, then

Question

If $ x=1 $ is a critical point of the function $ f(x)=\left(3 x^{2}+a x-2-a\right) e^{x} $, then, $ x=1 $ and $ x=-\frac{2}{3} $ are local minima of $ f $, $ x=1 $ and $ x=-\frac{2}{3} $ is a local maxima of $ f $, $ x=1 $ is a local maxima and $ x=-\frac{2}{2} $ is a local minima of $ f $, $ x=1 $ is a local minima and $ x=-\frac{2}{3} $ are local maxima of $ f $

MARKS App · Accepted Answer

f ( x ) = 3 x 2 + a x - 2 - a e x f ' ( x ) = 3 x 2 + a x - 2 - a e x + e x ( 6 x + a ) = e x 3 x 2 + ( a + 6 ) x - 2 ∵ x = 1 is a critical point ∴ f ' ( 1 ) = 0 ∴ 3 + a + 6 - 2 = 0 a = - 7 ∴ f ' ( x ) = e x 3 x 2 - x - 2 = e x 3 x 2 - 3 x + 2 x - 2 = e x ( 3 x + 2 ) ( x - 1 ) ∴ maxima at x = - 2 3 ∴ minima at x = 1

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