Question

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?

Answer 1

No, there cannot be a minimum product. The product of two numbers is maximized when the numbers are equal or close to each other.

Step-by-step explanation:

The student is asking if there can be a minimum product when x and y are numbers whose sum is 50 and we are trying to maximize the product xy. The answer is no, there cannot be a minimum product. This is because the product of two numbers is maximized when the numbers are equal or close to each other. In this case, when x = 25 and y = 25, the product xy is maximized at 625. If x and y were further apart, the product would be smaller.

Thus,

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

Answer 2

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.