Difference between revisions of "022 Exam 2 Sample B, Problem 8"
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−  '''Find the Maximum:''' The equation for  +  '''Find the Maximum:''' The equation for profit at a given production level is 
   
Latest revision as of 17:32, 17 May 2015
Find the quantity that produces maximum profit, given demand function and cost function
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 

and 

Using these equations, we can find the maximizing production level by determining when the first derivative of profit is zero. 
Solution:
Step 1: 

Find the Profit Function: We have 

From this, 

Step 2: 

Find the Maximum: The equation for profit at a given production level is 

Applying our power rule to each term, we find 

The only root of this occurs at , and this is our production level to achieve maximum profit. 
Final Answer: 

Maximum profit occurs when we produce 10 items. 