Can anyone help me solving these questions? with steps please.

Question

Take the function f(x) = x² e^-x,

a) Determine the intervals of increase and decrease for f(x).

b) Determine the intervals of concavity for f(x).

c) Determine the coordinates of any points of inflection.

Answer 1

f(x) = x²e^x

f'(x) = (x²+2x)e^x

set f'(x) = 0 for critical points: x² = -2x x = 0, -2

test in three regions to determine if slope is + or - x<0, 0<x<2, and x>2

Look at second derivative sign (+ is concave up) (- is concave down):

f''(x) = (x²+4x+2)e^x

f''(x) = 0 when x²+4x+2 = 0 x = (-4 +/- sqrt(16-8))/2 = -2 +/- sqrt(2) (both points of inflection. coordinates are (x, fIx))

find f''(critical points) to determine curvature at crit points and whether min, max.