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I am having trouble with a maximum problem and I'm not quite sure where I am going wrong so I will type of the problem and what I have done so far.

Find two positive numbers whose sum is 18 and the product of the first number and the square of the other is a maximum.

Here is what I've done so far:

x + y = 18 ---> y = 18 - x

xy^2=P

P = x(18-x)^2

P = x(324 - 36x + x^2)

P= x^3 - 36x^2 + 324X

To find where there is a maximum I found the first derivative of the equation above:

dP/dx = 3x^2 - 72x + 324

This is where I'm stuck, I know I want to make the first derivative equal to zero so I can find the values for the maximum, and verify my answer using the second derivative, but the first derivative cannot be factored. I must be doing something terribly wrong. The first time I did it I got x = 18 and y = 0 which cannot be right, to get that answer I took out a common x value in the equation P equation, but when I did the second derivative test it showed that the answer was actually a minimum. If anyone could give me some direction here I would really appreciate it.

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# Homework Help: Max and Min Problem

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