The closed interval method is a way to solve a problem within a specific interval of a function. The solutions found by the closed interval method will be at the absolute maximum or minimum points on the interval, which can either be at the endpoints or at critical points. Other ways of solving optimization problems include using the first derivative test or the second derivative test.
A really important part of Calculus is solving optimization problems so I want to explain to you what that is first. Let's take a look at a graph of a function y equals f of x and here I've pointed out some important key points on the graph but notice that my graph sort of terminates on the left end point a and right end point b. This is a function that's defined over a closed bounded interval a b what that means is closed means that it contains its end point. The function is going to be defined at each end point and bounded means that it's got a limited length. So this is a function defined over a closed bound interval all such functions if they're continuous they're going to achieve an absolute maximum value right that's our highest point on the graph and an absolute minimum value somewhere and the key is where do they achieve these values? They achieve the values either at an end point or at a critical point.
An optimization problem is all about finding absolute maxima and minima and in this first method that we study called the close interval method we're going to look at end points and critical points. These are the only places that you need to look when you're looking for an absolute max or min for a function of a closed bounded interval. So here is the closed interval method all spelled out, suppose f is a continuous function on some closed bounded interval a b find the critical point, so that's the first step, find all the critical points for the function f on that interval. Second evaluate f at the end points and at all the critical points. The largest of these values is going to be the absolute maximum and the smallest will be the absolute minimum and that's it. It's a 3 step process it's very easy to use and this is what we're going to use in our first few problems of optimization.