Let a be a positive constant and define f by f(x)=(a^2)(x^2)-(x^4). find the maximum and minimum values of f on [0,2a].

f(x)=(a^2)(x^2)-(x^4)

(f^1)(x) = (a^2)(2x)-4x^3

= 2(a^2)x - 4x^3

= 2x((a^2)-2x^2)

2x=0 (a^2)-2x^2 =0

x=0 2x^2 = a^2

x^2=(a^2)/2

x= a/sqrt2

The critical points are 0, 2a, and a/sqrt2

To find the min and the max values we would substitute the critical points in f(x). Is this right so far?