Minimize the sum of two numbers, subject to a condition on their product: The product of two numbers is 72. The sum of one number and 2 times the second is to be minimized. Determine the two numbers.

We need variables for the two numbers, a and b. The first condition can be stated as

.

The condition on the sum, S, can be stated as

.

To express the minimum condition, we need to express S in terms of one variable and differentiate. We can solve the first equation to get

. We can substitute this into the expression for S :

.

Now we can differentiate to get the derivative for any value of b:

.

To find the minimum at b_min : , or , or

Now we can use this result in the definition of a to find the corresponding value of a:

So the two numbers are 6 and 12. As a check, we can calculate S for several pairs of nearby values: