Solving optimization problems is one of the most important applications of Calculus towards real–life situations. Optimization problems specially employ one’s understanding about the derivatives of a function. More so, if these real–life situations can best be summarized into a mathematical model in a form of an equation or function in terms of the variables involved. In most of these problems, especially those concerning Economics, knowledge about finding the lowest and highest values of a function is very important. For instance, an Economist would like to approximate the amount of cost that will maximize profit or an Engineer would probably want to know the dimensions of a space figure that maximize its volume/capacity. These objectives can best be attained by applying the concept of optimization.

 Optimization problems involve the finding of the highest and the lowest values of a function within a certain interval. These values are collectively known as extrema. Extremum is the singular term. The highest point is called the maximum. The plural term is maxima. The lowest point of a function is commonly known as the minimum. The plural term is minima. To locate the extrema of a function on a closed interval, do the following steps:

  1. Find the first derivative of the given function using previously known derivative rules.
  2. Set the derivative function equal to zero. Solve for the value(s) of the independent variable x using previously discussed algebraic methods.
  3. Double check and make sure if the values obtained in step 2 are within the given closed interval.
  4. Make a table of values using the results obtained in step 2 and all the points between them. You can check as well on the endpoints of the given interval, in case the results in step 2 are different numbers.  Plug in these values into the original function.

Compare all the values you obtained in step 4. Identify the highest and the lowest of these values. These values are called the extrema of the function. The highest value of f(x) is the maximum of the function while the lowest value of f(x) is the minimum.

Example Max and Min on a Closed Interval Problem