We want to maximize Q = x*y^2 such that x + y^2 = 10, and x and y are positive.
We will find that the maximum value of Q is 25.
We start with our equation and restriction:
Q = x*y^2
x + y^2 = 10
Now we can isolate one variable in the restriction to get:
x = 10 - y^2
Now let's replace that in our equation:
Q = (10 - y^2)*y^2 = 10*y^2 - y^4
Now we need to derivate with respect to y to find the maximums and minimums, we will get:
Q' = 2*10*y - 4*y^3 = 20*y - 4*y^3 = y*(20 - 4*y^2)
The maximums and minimums happen when Q' = 0, then we need to solve:
Q' = y*(20 - 4*y^2) = 0
One solution is trivial, and is y = 0, the other solutions come from making zero the parentheses, we get:
20 - 4*y^2 = 0
y^2 = 20/4 = 5
y = ±√5
Notice that x and y must be positive numbers, so we take the only positive solution that we got, this is:
y = √5
Now we can get the value of x as:
x = 10 - y^2 = 10 - √5^2 = 5
Then the maximum value of Q is given by:
Q = 5*√5^2 = 25
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