I really need help with these optimization problems. Thanks.

Question

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.

Answer 1

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.

Answer 2

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.