If you are looking for where the break-even points are, you must determine the quantity for which

revenue = cost

Alternately, you may find the profit by calculating

profit = revenue – cost

The problem below demonstrate these strategies starting from a demand function and a cost function. To apply either of the relationships above, you need to form the revenue function from

revenue = (price)(quantity)

where the price is given by the demand function and Q represents the quantity.

**Problem** The demand function for *Q* units of a product is given by

The cost function is given by the function

a. Find the revenue function *R*(*Q*).

b. Find the break-even point(s)?

c. On a graph of *R*(*Q*) and *C*(*Q*), where do the break-even points lie?

d. Find the profit function *P*(*Q*).

e. Where do the break-even points lie on the graph of *P*(*Q*)?

**Solution 1 **To find the break-even point, this group of students set *R*(*Q*) = *C*(*Q*). This results in a quadratic equation. They moved all terms to one side and used the quadratic formula to find the quantities at which the revenue is equal to the cost.

**Solution 2 **This group of students found the profit function *P*(*Q*) first. Then they set it equal to zero to find the break-even points. Like the first solution, they also needed to use the quadratic formula.

Both techniques lead to the same break-even points and are equally valid. The only thing the second solution left out was the graph of the profit function showing the break-even points at the zeros (horizontal intercepts) of the function.