022 Sample Final A, Problem 9
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Given demand , and cost , find:
 a) Marginal revenue when x = 7 units.
 b) The quantity (xvalue) that produces minimum average cost.
 c) Maximum profit (find both the xvalue and the profit itself).
Foundations: 

Recall that the demand function, , relates the price per unit to the number of units sold, .
Moreover, we have several important important functions: 

In particular, we have the relations 

while 

and 

The marginal profit at units is defined to be the effective profit of the next unit produced, and is precisely . Similarly, the marginal revenue or marginal cost would be or , respectively.
On the other hand, any time they speak of minimizing or maximizing, we need to find a local extrema. These occur when the first derivative is zero. 
Solution:
(a): 

The revenue function is
Thus, the marginal revenue at a production level of units is simply 
(b): 

We have that the average cost function is
Our first derivative is then This has a single positive root at , which will correspond to the minimum average cost. 
(c): 

First, we find the equation for profit. Using part of (a), we have
To find the maximum value, we need to find a root of the derivative: which has a root at . Plugging this into our function for profit, we have 
Final Answer: 




Note that monetary units were not provided in the statement of the problem. 