Two numbers, x and y, add up to 10. What is the largest their product could be?

Question

I don't know how to do this problem, help.

Answer 1

Given:  x + y = 10

 

Maximize:  P = xy

 

Since x + y = 10, y = 10 - x

 

Therefore, P = x(10 - x)

 

                     = -x² + 10x

 

P' = -2x + 10

 

P' = 0 when x = 5  There is a max when x = 5 because the graph of P is a parabola opening downward with vertex when x = 5.

 

When x = 5, y = 10 - 5 = 5

 

Maximum product = (5)(5) = 25