Which are the two numbers whose sum is 20 and the product is the maximum.

Let us assume one of the numbers is x. As the sum of the
two numbers is 20, the other number is 20-x

The product of
the two numbers is a function f(x) = x*(20-x) = 20x-
x^2.

Now we need to find the maximum value of f (x). For
this we need the derivative of f(x) and have to equate it to
0.

f’(x) = 20 – 2x =
0

=> 10 – x
=0

=> x =10.

Now f’’(x)
= -2 which is negative at x=10. So f (10) is truly the maximum
value.

If x=10, the first number is 10 and the second
number is 20-10 = 10.

So the two required
numbers are 10 and 10.