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.
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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.
by Level 10 User (57.7k 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.

by Top Rated User (600k points)

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