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How to Find the Maximum Profit in Calculus

In a realistic application, profit maximization equations are used to determine how many units need to be produced to obtain the greatest profit returns. Unlike in calculus where the equations of the cost and revenue are given to you, companies must derive their own complex equations to find the maximum profit. With the cost and revenue equations provided in the calculus problem, you can figure out the maximum profit with a few simple calculations.

Things You'll Need

Caluclator

Instructions

- 1
  Locate the cost and revenue functions. When solving the maximize profit in calculus, the problem will generally provide you with the cost and revenue function to start off, but will ask you to solve for "x." In a maximize profit problem, the "x" represents the number of units you must produce to generate the most profit.
- 2
  Plug your cost and revenue functions into the maximize profit equation: P(x) = R(x) - C(x) where "R(x)" is the revenue function and "C(x)" is the cost function. For example, if your cost function is C(x) = -15x + 10 and your revenue function is R(x) = .10x^2 + 2x, then your equation would be:
  
  P(x) = (.10x^2 + 2x) - (-15x + 10)
- 3
  Simplify the maximize profit equation you found in Step 2. For example, if you take the equation P(x) = (.10x^2 + 2x) - (-15x + 10) and simplified it, it would look like this:
  
  P(x) = .10x^2 - 17x - 10
- 4
  Take the derivative of the simplified equation and set it to zero in order to solve for "x." For example, if our equation was P(x) = .10x^2 - 17x - 10, the derivative set to zero would be:
  
  0 = .20x - 17
- 5
  Find the number of units you will have to produce to maximize the profit by solving for "x." For example, if the derivative of our equation is 0 = .20x - 17, you would need to produce 85 units to create a maximum profit.

Numerology