Extrema on an Interval Where is the highest, where is the lowest point on a curve?The highest point on a curve is called the peak. The lowest point on a curve is called thetrough. The highest and lowest points are also called the extrema of a curve, or simplyextremas. Let's talk about extrema, maximums and minimums. We've got a graph here of f. So the smallest value attained on the graph of f, well, looking at these here, and we shouldprobably label. These are counting by ones. So 3, 4, 5, 6, 7, 8, 9, all the way up to 10 it lookslike. Negative one, negative two, one, two, so on.Anyway, it looks here that the smallest value, the lowest, y value on the graph of f occurs here atfour . Let's look at the graph of f and discuss how it behaves as x approaches infinity. The smallestsolution of the equation is -3, which occurs at x = 4. This is called the global minimum value.So if you ever get a what is, that's going to be your x, and if you get a where is, that's goingto be your y. The largest value attained will occur right here according to this equation.The maximum occurs at x equals zero, and this is known as the global maximum. The words minimum and maximum taken together are known as extrema . We have a continuous function f, which is graphed in the figure above. There are specificlocations where the function's maximum occurs, as shown in the graph. In this case, themaximum occurred at 0 and 10.It is important to note that we are working with a closed interval. The first event occurred atthe beginning of this interval, and the second event occurred at a specific point in between.The second event occurred at a point where the graph changed directions. Now how do weknow how the graph would change directions? Well, this is what is known as a critical pointor a critical number. It's a place where the derivative is equal to zero or is not defined.The relative maximum or minimum of this function occurs at this point. The location is aspecial point for the graph because it is a relative maximum or minimum. This is thebeginning of the interval. The function could have also ended here, but it did not happen inthis case.
Every closed interval of real numbers, a ≤ x ≤ b, has at least one maximum and one minimumvalue. These will occur at critical points of the function or at the endpoints of the interval. This isknown as the Extreme Value Theorem. Now, let's look at an interval that's not closed. Let's take the graph of f and add arrows to itsends. This will show us what the smallest value attained on the graph of f is.This seems to be lowest point on the curve. It occurs at the x value of four and the y value ofnegative three. Now, when we approach the largest value on this curve, tracing it to its endpoints, we see that it tends to infinity in both directions. The graph does not approach a largest value because the y value increases without bound.As a result, this function never truly reaches an endpoint. The limit to both positive andnegative infinity is positive infinity here. And so what this tells us is that an open interval is not guaranteedto have a maximum or a minimum. An absolute minimum or an absolute maximum occurs ata critical point and an endpoint as long as the interval of convergence is closed off.