When the product of two positive numbers is 88 what is the maximum sum of their squares
The product of two numbers is 88. Let one of the numbers be X, the other number is `88/X` . The sum of the square of the two numbers is given by the expression, `S = X^2 + (88/X)^2`
To determine the value of X that maximizes the sum, determine the derivative of S with respect to X.
`(dS)/(dX) = 2X - 15488/X^3`
Solve `(dS)/(dX) = 0`
=> `2X - 15488/X^3 = 0`
=> `X^4 = 7744`
=> `X = 2*sqrt 22`
`S'' = (2X^4+46464)/X^4`
At `X = 2*sqrt 22` , S'' is positive. This indicates that the value of S at `X = 2*sqrt 22` is minimum. The maximum value of the sum of the squares is infinity.
The two numbers with product 88 are X and `88/X` , as X takes on a smaller value, the value of `88/X` increases with a limiting case of 0 and infinity for X and `88/X` respectively. This makes the maximum value of the sum of the square of the two numbers equal to infinity.