How do you find two real numbers whose difference is 40 and whose product is a minimum?

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Answer 1 · 2016-10-25T18:30:35

Let the numbers be #x# and #y#, with #y > x#.

#y - x = 40#

#y = 40 + x#

Let #P# be the product.

#P = xy#

#P = x(40 + x)#

#P = x^2 + 40x#

We can determine the minimum product by completing the square and finding the coordinates of the vertex.

#P = 1(x^2 + 40x + 400 - 400) #

#P = 1(x^2 + 40x + 400) - 400#

#P = 1(x + 20)^2 - 400#

So, the minimum product is #-400# and the two numbers that would give this product are #20# and #-20#.

Hopefully this helps!