Solving Optimization Problems

  1. Read the problem.
  2. Draw a picture of the scenario, if you can.
  3. Label the lengths or variables in the picture and any variables in the problem.
  4. Reread the problem and write any extra information in the problem in terms of the variables.
  5. Decide what you are trying to optimize (maximize or minimize). Write it as a function of your variables. We'll refer to this function as f, even though you may have named it something else.
  6. Using the relationships between your variables from the extra information in the problem, rewrite the function to be optimized in terms of just one variable.
  7. Take the derivative of f.
  8. Set the derivative of f equal to 0. We're doing this, because at the relative maxima and minima of f, the derivative is 0. So if we find where the derivative of f is 0, we'll definitely find these points.
  9. Solve for the variable in the equation f'=0. Also, find the values of the variable for which f' does not exist. All of these points---where the derivative is 0 and where it doesn't exist---are called critical points.
  10. Find f''.
  11. Check each zero of the derivative to see if f'' is positive or negative there. If f'' is positive, the function is concave up at this point, so this zero of f' corresponds to a relative minimum. If f'' is negative, the function is concave down at this point, so this zero of f' corresponds to a relative maximum. If f'' does not exist (as it will not at each point for which f' doesn't exist) or is equal to 0, look at f' to see if the function was increasing or decreasing before and after the zero. This information will tell you which critical points are relative maxima and minima.
  12. If the variable was in a bounded range (say it couldn't be less than 0 or greater than 100), then look at the value of the function at each critical point of interest---the relative maxima, if you are trying to maximize; the relative minima, if you are trying to minimize---and at the end points of the range (0 and 100), and decide which value optimizes the function.
  13. Reread the question to make sure you give the answer requested.