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Find two nonnegative numbers whose sum is 9 and so that the product of one
number and the square of the other number is a maximum 10.
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Formulation of your post is NOT CORRECT and makes a few sense, if any . . .
Meanwhile, an interesting problem may stay behind it --- so I edited it, to make sense and to create really interesting problem.
My editing is as follows.
Prove that if the sum of two non-negative real number "x" and "y" is 9, x + y = 9,
then the product has the local maximum value of 108.
Find the values of x and y that provide this maximum.
With this reformulation, the solution is below.
Solution
If x+y = 9, then y = 9-x, and the function f(x,y) = is
f(x,y) = = .
So, we need to find the maximum of the function of x g(x) = .
To find the maximum of the function g(x), take its derivative and equate it to zero.
The derivative is
g'(x) = - 2x*(9-x) = - = .
Equating it to zero, you get
= 0, which is equivalent to = 0.
Left side is factorable
(x-9)*(x-3) = 0.
and two solution of the quadratic equation are x= 9 and x= 3.
The value x= 9 provides the local minimum of the function f(x), while x= 3 provides the local maximum.
At x= 3, y= 9-3 = 6, and the function g(x,y) = = 108.
Plot g(x) = x*(9-x)^2
Solved.