A line goes through the origin and a point on the curve Y x

A line goes through the origin and a point on the curve Y = x2e-3x, for x 0. Find the maximum slope of such a line. At what x-values does it occur?

(x, y(x)) is a point on the curve y = x^2e^(-3x).

Find the slope of the line through (0, 0) and (x, y(x)):

m = (y(x) - 0)/(x - 0) = y(x)/x = (x^2e^(-3x))/x = xe^(-3x)

Maximize m:

m\' = e^(-3x) - 3xe^(-3x) = e^(-3x)(1 - 3x)

m\' = 0 => 1 - 3x = 0 => x = 1/3

Maximum slope: m(1/3) = (1/3)e^(-1)