Maximize Profit: Let the Cost of production, C, depend on the number of units, x, according to the equation

.

Let the number of units, x, depend on the price per unit, p, as

Find the price per unit at which the maximum profit, P, will be realized.

Since P = R - C, where R = revenue = (price per unit) * (number of units) = p*x,

We need to express the profit in terms of price, p, alone. That is, we need to replace x in our equations for R and C. Using the given equation, x = 80 - p, we find

, and

Now we combine these to express the profit in terms of price per unit alone:

, or

Now we can maximize by taking the derivative

. Setting this to 0 we get p = 60.

We can check this result by calculating the profit for the values of p around this, say 59, 60, and 61. The resulting values of the profit are 249, 250, and 249 respectively. Hence p = 60 does maximize the profit.