Look for critical points (using the derivative). Identify which ones (if any) are maximum values.

Check to see if there is one maximum inside the interval.

If no, then the maximum will be at one end or the other.

If yes, check to see if it's value (at the maximum inside the interval) is higher than the value at the highest border point.

plot and look at the graph...x=3

Potential solutions are the values of the function at the interval boundary and any relative extrema within the interval.

Boundary Values:

f(-1) = 1/2 + 4/3 + 2 = (3+8+12)/6 = 23/6

f(3) = 81/2 - 36 + 2 = (81 - 72 + 4)/2 = 13/2

Relative Extrema:

f'(x) = 2x³ - 4x² = 2x²(x-2) = 0

x = 0, 2

x = 0 is an inflection point and can be discarded

x = 2 is within the interval

f(2) = 8 - 32/3 + 2 = (24-32 + 6)/3 = -2/3

Set of potential solutions = {-2/3, 23/6, 13/2}

Ans: Maximum value on the interval is 13/2 at x = 3