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Calculus Optimization Problem Solving Strategy

We will use the steps outlined below to solve each Calculus Optimization problem on this site, step-by-step, every single time. We hope that this will help you see the strategy we’re using so you can learn it too, and then be able to apply it to all of your problems, especially those on your exams. That’s as opposed to learning just how to solve a particular problem on your homework, say, since you may well never see that specific problem again.

The strategy consists of two Big Stages. The first does not involve Calculus at all; the second is identical to what you did for max/min problems.

Stage I: Develop the function.

Your first job is to develop a function that represents the quantity you want to optimize. It can depend on only one variable. The steps:

  1. Draw a picture of the physical situation.
    Also note any physical restrictions determined by the physical situation.
  2. Write an equation that relates the quantity you want to optimize in terms of the relevant variables.
  3. If necessary, use other given information to rewrite your equation in terms of a single variable.

Stage II: Maximize or minimize the function.

You now have a standard max/min problem to solve.

  1. Take the derivative of your equation with respect to your single variable. Then find the critical points.
  2. Determine the maxima and minima as necessary.
    Remember to check the endpoints if there are any.
  3. Justify your maxima or minima either by reasoning about the physical situation, or with the first derivative test, or with the second derivative test.
  4. Finally, check to make sure you have answered the question as asked: Re-read the problem and verify that you are providing the value(s) requested: an x or y value; or coordinates; or a maximum area; or a shortest time; whatever was asked.
To see this strategy in action, please see our Optimization page.

Update September 2022

All of our content is now free, with the goal of supporting anyone who is working to learn Calculus well.

To access our materials, please simply visit our Calculus Home screen.

Also new: we've added a forum,, also free to use.
If you need some help with a Calculus question, please post there and we'll do our best to assist!