The revenue when selling x items of a certain product is? (x) = 40x - x^2. The cost of producing x items is C (x) = 2x^2 + 4x + 10. Find the number of items you should sell to maximize the profit? Also determine the maximum profit. (Hint: Profit is Revenue minus Cost, so P (x) = R (x) - C (x))

Step-by-step explanation:

First thing we need to do is to find the profit function. We will do that by the subtraction that was provided to us in the hint.

P (x) = 40x - x² - [2x² + 4x + 10] which simplifies to

P (x) = - 3x² + 36x - 10

From this, we can determine that the function is parabolic (quadratic) and that it has a maximum value. The negative sign in front of the function tells us that this is an upside down parabola that has a max value located at its vertex. It is within the vertex that we will find the information we need. In order to determine the vertex, we need to put this quadratic into vertex form. We do this by completing the square.

The first rule of completing the square is to set the quadratic equal to 0 and then move over the constant to the other side of the equals sign:

-3x² + 36x = 10

The next rule is that coefficient on the x-squared term, the leading coefficient, is a 1. Ours is a - 3, so we have to factor it out:

-3 (x² - 12x) = 10

Next is the process that will complete the square on the quadratic and create a perfect square binomial we need for the h coordinate of the vertex. We take half the linear term, square it, and add it to both sides. Our linear term is 12. Half of 12 is 6, and 6 squared is 36, so we add it first into the parenthesis. BUT we cannot disregard the - 3 sitting out front. It is a multiplier. That means that whatever we add into the parenthesis on the left is added in as - 3 (36) on the right:

-3 (x² - 12x + 36) = 10 - 108

Writing the left side in its perfect square binomial and at the same time simplifying the right:

-3 (x - 6) ² = - 98

Almost done. Now we move the - 98 back over by addition, and set the quadratic back equal to y:

-3 (x - 6) ² + 98 = y

Our vertex is apparent now at (6, 98). In the context of our problem, the h value is the number of items needed to produce the max profit of k. 6 items need to be sold in order to produce a max profit of $98.