For what values of x do the curves y=(1+x^3)^2 and y=2x^6 have the same slope?
y1 = (1 + x^3)^2
y2 = 2x^6
d y1/dx = 2(1 + x^3)(3x^2)
d y2/dx = 12x^5
we want
12x^5 = 6x^2(1 + x^3)
12x^5 = 6x^2 + 6x^5
6x^5 - 6x^2 = 0
6x^2 (x^3 - 1) = 0
x = 0 or x = 1