If x = 1 is a critical point of the function f ( x ) = 3 x 2 + a x - 2 - a e x , then

If x=1 is a critical point of the function f(x)=3x2+ax-2-aex, then 
  1. x=1 and x=-23 are local minima of f
  2. x=1 andx=-23 is a local maxima of f
  3. x=1 is a local maxima andx=-22 is a local minima of f
  4. x=1 is a local minima andx=-23 are local maxima of f

Solution

f(x)=3x2+ax-2-aex
f'(x)=3x2+ax-2-aex+ex(6x+a)=ex3x2+(a+6)x-2

∵  x=1 is a critical point   ∴  f'(1)=0

∴3+a+6-2=0
a=-7

∴f'(x)=ex3x2-x-2=ex3x2-3x+2x-2=ex(3x+2)(x-1)

∴ maxima at x=-2 3∴ minima at x=1

Asked in: JEE Main 2020 (05 Sep Shift 2)

Practice more Applications of Derivatives questions on Aicharya