Of all numbers x and y whose sum is 50, the two that have the maximum product are xequals=25 and yequals=25. that is, if xplus+yequals= 50, then xequals=25 and yequals=25 maximize qequals=xy. can there be a minimum product? why or why not?

x + y = 50

Q = x * y

Combining the two:

Q = x * (50 - x)

Q = 50x - x^2

Q = - x^2 + 50x

Q = - (x^2 - 50x)

Q = - (x^2 - 2 * 25x + 25^2 - 25^2)

Q = 25^2 - (x - 25) ^2

Q = 625 - (x - 25) ^2

So what we got is an equation for parabola with a vertex at (25, 625) and it opens downward. We know that parabolas only have one critical value, so if x and y are unrestricted, then there's no minimum product.