SOLUTION: Find two positive real numbers whose product is a maximum. (Enter your answers as a comma-separated list.) The sum of the first and twice the second is 16.

Let the second number be x.

Then the first number is 16-2x.


We want to find the numbers to get the maximum value of the product  x*(16-2x).


This function,  f(x) = x*(16-2x),  is a quadratic function  f(x) = 16x - 2x^2.


Since the coefficient at x^2 is negative, its plot is a parabola opened downward, and it really has the maximum.


The zeroes of this quadratic function are x= 0 and x= 16/2 = 8,  from its decomposition f(x) = x*(16-2x).


The maximum value is located midway between the roots 0 and 8;  so, the maximum is at x= 4  and  is equal to


    4*(16-2*4) = 4*8 = 32.


ANSWER.  The maximum value of the product of the numbers is 32.

         It is achieved when second number is  4  and first number is  16 - 2*4 = 8.