For what values of x is the graph of y=x e^{-4 x} concave down?

Answer 1

y = xe^(-4x)
y' = (1-4x)e^(-4x)
y" = 8(2x-1)e^(-4x)

since e^(-4x) > 0 for all x, y" < 0 when 2x-1 < 0.

see the graph at

http://www.wolframalpha.com/input/?i=+xe^%28-4x%29+for+0+%3C%3D+x+%3C%3D+1