Now we put our optimization skills to work.

An optimization problem is a problem where you need to maximize or minimize some quantity given some constraints. This can be accomplished using the tools of differential calculus that we have already developed.

Perhaps the most basic optimization problems is generated by the following question:

Among all rectangles of a fixed perimeter, which has the greatest area?

Let’s not do this problem in the abstract, let’s do it with numbers.

A key step is to explain why is actually the maximum. Above we basically used facts about the derivative. Below we use a similar argument.

Hence, calculus gives a reason for why a square is the rectangle with both

  • the largest area for a given perimeter.
  • the smallest perimeter for a given area.

We may be done with rectangles, but they aren’t done with us. Here is a problem where there are more constraints on the possible side lengths of the rectangle.

Again, note that above we used the Extreme Value Theorem to guarantee that we found the maximum.