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Calculus problem - optimization

Find two positive numbers whose sum is 9 and so that the product of one number and the square of the other number is a maximum.
What are we trying to maximize? The product P of one number x and square of the other number y:
With only this equation, the best way to maximize P is to let x and y be infinite. That choice of x and y does not meet all the requirements of the problem, however. We also have a constraint:
x = 9 - y
Substituting the constraint into the product equation:
P=(9-y)*y^2 = 9y^2- y^3
To maximize P, we will find the derivative and set it equal to zero:
dP/dy = 18y -3y^2= 0
18y - 3y^2=0
So, y = 0 or 6.
To determine the maximum, we need to check the value of the product P at both y = 0 and y = 6
If y = 0, then x = 9 (why?) and P = 0
If y = 6, then x = 3 and P = 108
So, the numbers that maximize xy^2 are x=3 and y=6.



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