Applications of Derivatives. Extreme Values (Part I)

Applications of Derivatives. Extreme Values (Part I)

The absolute maximum of a function is the largest value that the function can take on. The absolute minimum of a function is the smallest value that the function can take on. In general, it is not possible to determine an absolute maximum or minimum unless you know what the graph looks like at every point in the domain.

We're going to consider some applications of differentiation. First, we will look at extreme values of a given function and try to find ways to locate extreme values.

Consider such an example of a function f(x):

The graph of the function is shown in blue. How do we know the extreme values of this function? The extreme values of this function occur when, where is the valley, is a peak, and so on.

So these points are where the function attains its maximum value, or maximums. There are actually two types of maximums.

One is called the absolute maximum of a function f, which means that f(x) is equal to or greater than all other values of f on an interval I.

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We say that function f attains an absolute maximum value on interval I at the point a, if the value of f on this interval is bigger than or equal to the value of f at all other places on the interval.

Identically, we can define absolute minimum in the same way that that the function attains absolute minimum at b means f(b) is less than or equal to all other values of the function in the interval.

A local maximum occurs when a function has a maximum value in a region containing more than one point. This means that the maximum is only a maximum in a very small region around the point.

A function f attains a local maximum value at an interior point c in the interval means that f(c) is greater than or equal to f(x), for all x in an open interval containing c.

And identically, for local minimum f(d) is less than or equal to f(x) for x in some open interval containing d. Local are characterized by the property that f(c) is big only in a small region around c. If we have absolute maximum, it's in all points in the interval.

Going back to the graph we saw just a few minutes ago, we see that here, the absolute maximum is at this point. This value is clearly the greatest of all values in the interval a b.

The minimum value in the interval a b is here, which is certainly the lowest value among all the values in that interval. This one is a local maximum, not an absolute maximum because there are larger values elsewhere in the interval but it's a local maximum because for points around him he is the biggest.

This is a local minimum, not an absolute minimum. However, it is the smallest value among the values around this point.

Now, even at this end point here, we can see that this is a local maximum because it's larger than any other point near him. If you are looking for an absolute maximum or absolute minimum, be careful to locate points far from other nearby points.

We must examine all the points in an interval, because we need to make sure that the maximum value is indeed the largest among all points in the interval.

How will we find the absolute maximum and minimum? Let us look at those examples.

This graph, the blue one, is the function we have. The interval under consideration is from minus infinity to infinity. What is the absolute maximum value of the function? The answer is no because you can get bigger and bigger values as you go further out in this interval. So there is no absolute maximum value for this function.

But there is a minimum point, at 0. This point is smaller than or equal to all other values. Now let us look at an example of this second concept.

Now, the interval under consideration is the closed interval from 0 to 1, compared with the previous example, you have just 0 and 1.

In this case, the graph of the function is this path. Obviously, the largest value occurs at this point, and the smallest value is here. So in this situation, the function has both an absolute maximum and an absolute minimum.

Now, let's consider another example. The graph looks very much like the previous two cases, but instead of being restricted to the interval from zero to one, this interval is greater than zero but less than or equal to one. The difference between this example and the previous one is that it does not contain zero.

Once again, we see that this one is the absolute maximum, but there is no absolute minimum. The point zero is not on the graph, so there is no absolute minimum. Another similar example is this one. Now, the interval under consideration is open between zero and one.

Now here, we do not have the absolute maximum nor minimum, because both zero and one are not on the graph. You can see from these four examples that although the graphs of the functions look very much alike, the interval or domain under consideration makes all the difference.