Question

Consider the function with rule f(x) = |x^2-ax|, where a is a positive constant.

Find the maximum value of the function in the interval [0,a]

How do you do find that?

Answer 1

if x^2-ax >= 0, |x^2-ax| = x^2-ax
That is, if x(x-a) >= 0
x <= 0 or x >= a
These intervals are outside the domain.

If x^2-ax < 0, then |x^2-ax| = -(x^2-ax)
That is, if x(x-a) < 0
So, for 0<=x<=a the maximum value of f(x) is at x = a/2

See a sample graph here:

http://www.wolframalpha.com/input/?i=%7Cx(x-2)%7C