SOLUTION: maximize q=5xy^2, where x and y are positive numbers such that x+y^2=8

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Question 1051078: maximize q=5xy^2, where x and y are positive numbers such that x+y^2=8
Found 2 solutions by Fombitz, ikleyn:
Answer by Fombitz(32388)   (Show Source): You can put this solution on YOUR website!

Substitute into q equation,

Now q is a function of one variable, y.
To find the maximum, differentiate with respect to y and set the derivative equal to zero.



So,


Remember y is positive,

So then,



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Answer by ikleyn(52787)   (Show Source): You can put this solution on YOUR website!
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maximize q=5xy^2, where x and y are positive numbers such that x+y^2=8
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The solution at the Algebra level, with no calculus. 




Substitute into q equation,



Now q is a function of one variable, y.

q = .

"Complete the square" :

q =  =  =  + .

It shows that q is maximal at   = 4,  or  y = +/-2.

According to the condition, only positive "y" are considered, so the solution is y=2.

Then x = 4.

The minimum of q is 80.


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