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 the trough. The highest and lowest points are also called the extrema of a curve, or simply extremas. Let’s talk about extrema, maximums and minimums. We’ve got a graph here of f. So the smallest value attained on the graph off, well, looking at these here, and we should probably label. These are counting by ones. So 3, 4, 5, 6, 7, 8, 9, all the way up to 10 it looks like. 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 at four Let’s look at the graph off and discuss how it behaves as ~ approaches infinity. The smallest solution 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 going to 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 specific locations where the function’s maximum occurs, as shown in the graph. In this case, the maximum occurred at 0 and 10. It is important to note that we are working with a closed interval. The first event occurred at the 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 we know how the graph would change directions? Well, this is what is known as a critical point or 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 special point for the graph because it is a relative maximum or minimum. This is the beginning of the interval. The function could have also ended here. but it did not happen 1n this case. [Graph] minimum+maximum=extrema 1) The smallest value attained on the graph of f is -3 2) This occurs at x=4 absolute/global minimum 3) The largest value attained on the graph of f is 4 4) This occurs at x=0 absolute/global maximum Every closed interval of real numbers, a<=x<=b, has at least one maximum and one minimum value. These will occur at critical points of the function or at the endpoints of the interval. This is known as the Extreme Value Theorem. Every closed interval [a, b] has an absolute max and absolut min. This will occur at a critical pt or endpoint. Now, let’s look at an interval that’s not closed. Let’s take the graph of f and add arrows to its ends. 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 of negative three. Now, when we approach the largest value on this curve, tracing it to its end points, 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 and negative infinity is positive infinity here. [Graph] 1) The smallest valuea attained on the graph of f is -3 2) This occurs at x=4 3) The largest value attained on the graph of f is O/ 4) This occurs at x= O/ An open interval is not guaranteed to have a max or a min And so what this tells us is that an open interval is not guaranteed to have a maximum or a minimum. An absolute minimum or an absolute maximum occurs at a critical point and an endpoint as long as the interval of convergence is closed off.
