Question

say you choose two positive numbers x,yx,y such that their sum is 100. what is the maximum value that their product xyxy could be? explain how you set the problem up and solve it, in addition to giving the answer.

Answer 1

The maximum value of the product is 2500 if the sum of two positive values, x and y, is 100.

The sum of two positive numbers x and y is 100.

x + y = 100

Subtracting y from each side of the equation.

x + y - y = 100 - y

x = 100 - y

The product of x and y is maximum.

Let P be the product.

Then,

P = xy

The product will be maximum when dP/dy = 0.

P = ( 100 - y )y

P = 100y - y²

Differentiating the equation.

dP/dy = 100 - 2y

100 - 2y = 0

Adding 2y on each side of the equation.

100 - 2y + 2y = 0 + 2y

2y = 100

Dividing each side of the equation by 2

y = 50

Then,

x + y = 100

x + 50 = 100

x = 50

Then, the product is:

P = xy = 50 × 50 = 2500

If the sum of two positive numbers x and y is 100, then the maximum value of the product is 2500.

Learn more about positive numbers here:

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