Find two positive numbers whose product is 144 and whose sum is a minimum? Optimization Help

Question

Find two positive numbers whose product is 144  and whose sum is a minimum.

Answer 1

The numbers are 12 and 12.

Consider the following two equations: x * y = 144, x + y = S.

Solving for y in the first equation and placing the result in the second equation we find: x + 144/x = S.

One takes derivative of S with respect to x and set the resulting expression to zero; 1 - 144/x^2 = 0.

Solving now for x we find x= 12. Placing this result in the first equation, we find y = 12.

This result can be understood graphically by plotting the function x + 144/x as a function of x, to verify that at the value of x=12, it reaches a minimum.

Answer 2

xy = 144

minimize x+y

x= 144/y

minimize 144/y + y

take the derivative and set equal to zero

-144/y^2 + 1 = 0

-144/y^2 = -1

y^2 = 144

y = sqr144 = 12

x = 144/12= 12

one number is 12 and so is the other. 24 is their minimum sum