What are the answers to both questions and what is the difference between the two?

The sum of two nonnegative numbers is 25. Find the two numbers so that the sum of the first plus the square of the second is a minimum.The sum of two nonnegative numbers is 25. Find the two numbers so that the sum of the first plus the square of the second is a maximum.
asked Feb 5, 2016 in Calculus Answers by anonymous

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2 Answers

Well the difference between them is even though they are the same numbers, you said negative and positive. I see -25 and +25 are different because -25 is below zero and +25 is above zero. That is what is different between them.
answered Feb 5, 2016 by Mathical Level 10 User (57,540 points)

Let the numbers be a and b. Then a+b=25, so b=25-a.

Also let y=a+(25-a)²=2a²-50a+625.

This can be written y=2(a²-25a+625/2)=2(a²-25a+625/4-625/4+625/2) = 2((a-25/2)²+625/4)

When a=b=25/2, y has a minimum value of 625/2=312.5.

When a=0, b=25, y has a maximum value of 625.

answered Feb 11 by Rod Top Rated User (582,400 points)

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