Question

find two mpositive real numbers whose product is a maximum and whose sum of the first number and four times the second is 240

Answer 1

A + 4B = 240
A = 240 - 4B
A*B = (240-4B)*B = -4B^2 + 240B
Use formula for finding line of symmetry of this parabola:
B = -240/(2*-4) = -240/-8 = 30
This value for B maximizes A*B.
By substitution
A = 240-(4*30) = 240 - 120 = 120
Therefore
A = 120, B = 30