How do you 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?

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Answer 1 · 2015-02-27T16:07:18

Let's call the first number #x#
Then the other will be #9-x# (as they add up to #9#

So the question translates to:
Find the maximum of #y=x^2(9-x)=9x^2-x^3#

The maximum is when the derivative is set to #0#

#y'=18x-3x^2=3x*(6-x)=0->x=6#
And the other number will be #9-6=3#

Check the answer: #6^2*3=108#

To be sure that this is a maximum you could check for:
#5.9 harr 3.1# and for #6.1 harr 2.9#