given x+y=25, x>0, y>0, Find the maximum value of xy

Answer 1

Hi!

From x+y=25 => y =25-x.

We want to find the max value of x*y = x*(25-x)

Note, x*(25-x) is a parabola concave down so it has a max value given by the y coordinate of its vertex.

To find its vertex, we write the equation into the vertex form by completing the square:

-x²+25x+(25/2)²-(25/2)²= -(x-(25/2)²)²+(25/2)²

Hence, its vertex is (25/2, 625/4). The max value is 625/4=156.25

Hope this helps! Please let me know if you have any questions. Good luck.

Answer 2

graph the line x+y = 25

it has x and y intercepts (0,25), (25,0)

connect them with a line segment, find the midpoint (12.5, 12.5)

then multiply the x and y coordinates to get 12.5(12.5) = 156.25 = max Area