Question

Find two positive real numbers whose product is a maximum. (enter your answers as a comma-separated list.) the sum of the first and three times the second is 54.

Answer 1

the sum of the first and three times the second is 54.

x + 3y = 54 

3y = 54 - x 

y=  54 /3  -  x  /  3  = 18 -x /3

xy = x  (18 -x /3) =  18x   -  x^2  / 3  =  - 1/3 x ^ 2  + 18 x 


since a<0, it is concave downward, and the vertex is the maximum value. That vertex occurs at x= - b /2a = (-18) / (-2/3) = 12


The first number is x=12.
The second is y = 18 -x /3 = 18- 12/ 3 = 18  - 4 = 14