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?

Damon Vazquez

Damon Vazquez

Answered question

2022-10-06

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?

Answer & Explanation

Kaitlyn Levine

Kaitlyn Levine

Beginner2022-10-07Added 12 answers

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 ) = 9 x 2 - x 3
The maximum is when the derivative is set to 0
y = 18 x - 3 x 2 = 3 x ( 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 3.1 and for 6.1 2.9

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