Introduction to Optimization Problems

#5.10 Introduction to Optimization Problems

We previously learned how to find the minimum or maximum value of a function on an interval using either the First Derivative Test or the Second Derivative Test. What significance does figuring out these pieces of information have? Well, they can help us solve optimization problems!

#👍 Optimization Problems

You may ask, what are optimization problems? If we think about what the word “optimizing” means, we see that these types of problems involve finding the best possible solution, often looking to maximize or minimize a certain quantity.

So how can we use calculus to solve these kinds of problems? Well, if we go back to its definition, we see that we want to “look to maximize or minimize a certain quantity.” Do we know how to find the minimum or maximum value of a function? We certainly do! How? By taking the derivatives of the function (applying either the First Derivative or Second Derivative Tests)!

One tricky part about optimization problems is that they usually add one more variable that we have to account for. But don’t fear, this just means you have to do the extra step of finding a relationship between two of the variables so that you can “get rid of” one through substitution.

#✏️ Optimization Walkthrough

Let’s go through a problem below to get a better idea of how to solve this type of problem:

Let $s = x^2 + y^2$. If $xy = -36$, what $x$ and $y$ values minimize $s$?

Since there are two variables, $x$ and $y$, and we do not know how to differentiate with respect to more than one variable, we should look to re-wri...