[FONT='Cambria','serif']There are 2 positive numbers who have a sum of 9. Determine the maximum value of the product of one number and the square of the other number.[/FONT]
Let be the function representing the product as stated WLOG:
so, and we want the maximum, thus we need such that
therefore, either or but note that x is a positive number, therefore we take .
optional: if , then we are sure that gives the maximum value for
and at , we have .