What is the maximum product of x and y

Question

Let x and y be two real numbers. The sum of x and 4y is 144. Find x and y such that the product of them is the largest. What is the maximum product of these two numbers?

Answer 1

x + 4y = 144 ---> x = 144 - 4y

x*y = (144 - 4y)*y = P(y) is the function

for the product in terms of y

Then P(y) = (144 - 4y)y = 144y - 4y^2

maximizing:

0 = dP/dy = 144 - 8y

So then

8y=144 ---> y =144/8 =18

then x = 144 - 4*18 = 144 - 72 = 72

the max product is 18*72 = 1296

Answer 2

4x+y= 144

y=144-4x

xy= x(144-4x) = 144x -4x^2

take the derivative and set equal to zero, then solve for x

144-8x =0

8x = 144

x = 144/8 = 18

y = 144-4(18) = 144-72 = 72

the numbers are 18 and 72

their product is 18(72) = 1440-144= 1296