Question
Consider the function f(x) = x^2/ e^x Find the critical points of this function and list it's intervals of inctrease and decrease. Justify your answer completely using Calculus I methods. Critical Point(s): f(x) is increasing on: f(x) is decreasing on: ___________________________________________________________________________________________ What I got: f '(x) = -e^(-x)(x-2)x f '(x) = 0 @ x=0, x=2 Critical Point(s): (0,0), (2, 4/e^2) f(x) is increasing on: (2-sqrt2, 2+sqrt2) f(x) is decreasing on: (-infinity, 2-sqrt2) ; (2+sqrt2, +infinity) ___________________________________________________________________________________________
Explanation / Answer
f(x) = x^2/e^x f'(x) = (2x-x^2)/e^x f'(x) = 0 at x =0 and x = 2 , hence 0 and 2 are critical points for f(x) to be increasing, f'(x) should be > 0 hence 2x - x^2 > 0 x^2< 2x hence increasing on (0,2) for it to decrease f'(x) < 0 hence x^2 > 2x therefore x belongs to ( 2, infinity) U ( -infinity , -2)