Thread: finding the max and min values

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?

Yes, it is. Since the problem is asking for the maximum and minimum values, the "global" max and min, you need only evaluate the function at those three points to find which gives the largest value of x and which the smallest.