Maximize the product of two numbers, subject to a condition on their sum: The sum of a number and 3 times another number is 36. The numbers are to be selected so that their product is a maximum. Determine the two numbers.

To solve such a problem, we need variables for each number, say a and b. The first condition can be expressed as

.

The second condition is P = a * b, where P is the unknown product.

To maximize this, we need to express P in terms of one variable and differentiate. From the first equation, we can solve for

a = 36 - 3 * b.

We can substitute this for a in the product:

Now we can differentiate and set the derivative to zero to find the maximum. First we get the derivative for any value of b:

.

To find the maximum at b_max:

_, or b_max = 6.

We can use this in the above equation to find the corresponding value of a_max:

So the two numbers are 6 and 18, with the maximum product of 6 * 18 = 108.

As a check, we can calculate the product for other pairs which satisfy the original condition, such as 5,21 and 7, 15. In both cases the product, 105, is smaller.