How do you find the two positive real numbers whose sum is 40 and whose product is a maximum?

1 Answer
georgef
Dec 10, 2016

You have to find first a function to represent the problem stated, and then find a maximum of that function

Explanation:

The problem states that we are looking for two numbers #x# and #y# such as #x+y=40#, that is

#y=40-x#

We would like to find where the product #x*y# is maximum, but from the above equation we can write:

#x*y=x*(40-x) = -x^2+40x#.

So we now have a one-variable function #f(x)=-x^2+40x#, and must find a positive value of #x# where the function #f# reaches a maximum.

To do that we calculate the derivative #f'(x)=-2x+40#, and we look for values of #x# where #f'(x)=-2x+40=0#. There is only one such value (critical point) with #x=20#.

Now the second derivative #f''(x)=-2# is negative everywhere, and therefore is negative at the critical point #x=20#. Hence, #x=20# is a maximum for #f#.

But we also know that #y=40-x#, so the value of #y# is also #20#.

The solution is then #x=20, y=20#

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