SOLUTION: Given that x and y are positive real numbers such that x^2-2x+4y^2=0, btain the maximum value of the product xy.

Question 1104622: Given that x and y are positive real numbers such that x^2-2x+4y^2=0, btain the maximum value of the product xy.
Answer by greenestamps(13203) (Show Source): You can put this solution on YOUR website!

Solve the quadratic for either x or y and substitute into the product xy to get the product as the function of a single variable. Then differentiate that product and find where the derivative is zero.

[since the problem says x and y are both positive]

Then the product xy is
[note: when I have to take a derivative like this, I find it easier to move the "x" inside the radical, so that I don't have to use the product rule when doing the differentiation.]

The derivative is

The derivative is zero when

When x = 3/2,

And at that point the product xy is

The maximum value of xy, given , is

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