Given any quantity described by a function, we are often interested in the largest and/or smallest values that quantity attains. For instance, if a function describes the speed of an object, it seems reasonable to want to know the fastest/slowest the object traveled. If a function describes the value of a stock, we might want to know the highest/lowest values the stock attained over the past year. We call such values extreme values.
Consider Figure 3.1.3. The function displayed in Figure 3.1.3.(a) has a maximum, but no minimum, as the interval over which the function is defined is open. In Figure 3.1.3.(b), the function has a minimum, but no maximum; there is a discontinuity in the “natural” place for the maximum to occur. Finally, the function shown in Figure 3.1.3.(c)has both a maximum and a minimum; note that the function is continuous and the interval on which it is defined is closed.
The image shows the graph of a function with the appearance of a parabola that opens downward, with its vertex at . The domain for the function is , so the graph approaches, but does not reach, the points and . These points are shown with hollow dots, to illustrate the fact that they are not part of the graph.
The image shows the graph of a function with the appearance of a parabola that opens downward, with its vertex at . The domain for the function is , this time, the points and at the ends of the graph are included.
It is possible for discontinuous functions defined on an open interval to have both a maximum and minimum value, but we have just seen examples where they did not. On the other hand, continuous functions on a closed interval always have a maximum and minimum value.
This theorem states that has extreme values, but it does not offer any advice about how/where to find these values. The process can seem to be fairly easy, as the next example illustrates. After the example, we will draw on lessons learned to form a more general and powerful method for finding extreme values.
The image shows the graph of on the interval . Beginning at the point , the graph rises to a peak at , before falling to its minimum value at . The graph then climbs to the endpoint , where it reaches its maximum value.
The graph is drawn in such a way to draw attention to certain points. It certainly seems that the smallest -value is , found when . It also seems that the largest -value is , found at the endpoint of ,. We use the word seems, for by the graph alone we cannot be sure the smallest value is not less than . Since the problem asks for an approximation, we approximate the extreme values to be and .
Notice how the minimum value came at “the bottom of a hill,” and the maximum value came at an endpoint. Also note that while is not an extreme value, it would be if we narrowed our interval to . The idea that the point is the location of an extreme value for some interval is important, leading us to a definition of a relative maximum. In short, a “relative max” is a -value that’s the largest -value “nearby.”
The graph of is shown, for between and . Arrows at either end of the graph indicate that will approach in either direction beyond the interval shown. The graph shows three relative extrema: there are minima when and , and a maximum at .
We still do not have the tools to exactly find the relative extrema, but the graph does allow us to make reasonable approximations. It seems has relative minima at and , with values of and . It also seems that has a relative maximum at the point .
The figure implies that does not have any relative maxima, but has a relative minimum at . In fact, the graph suggests that not only is this point a relative minimum, is the minimum value of the function.
What can we learn from the previous two examples? We were able to visually approximate relative extrema, and at each such point, the derivative was either or it was not defined. This observation holds for all functions, leading to a definition and a theorem.
Be careful to understand that this theorem states “Relative extrema on open intervals occur at critical points.” It does not say “All critical numbers produce relative extrema.” For instance, consider . Since , it is straightforward to determine that is a critical number of . However, has no relative extrema, as illustrated in Figure 3.1.17.
The graph of rises steadily as increases. It briefly levels off at , where there is a horizontal tangent. The horizontal tangent corresponds to a critical point, but it is not a relative maximum nor a relative minimum.
Theorem 3.1.4 states that a continuous function on a closed interval will have both an absolute maximum and an absolute minimum. Common sense tells us “extrema occur either at the endpoints or somewhere in between.” It is easy to check for extrema at endpoints, but there are infinitely many points to check that are “in between.” Theorem 3.1.15 tells us we need only check at the critical points that are in between the endpoints. We combine these concepts to offer a strategy for finding extrema.
The image shows the graph of on the interval . There appears to be a relative minimum near , and no other critical points. The absolute maximum value of the function appears to occur at the right endpoint of the interval.
Next, we find the critical values of on .; therefore the critical values of are and . Since does not lie in the interval , we ignore it. Evaluating at the only critical number in our interval gives: .
Figure 3.1.21 gives evaluated at the “important” values in . We can easily see the maximum and minimum values of : the maximum value is and the minimum value is .
Note that all this was done without the aid of a graph; this work followed an analytic algorithm and did not depend on any visualization. Figure 3.1.20 shows and we can confirm our answer, but it is important to understand that these answers can be found without graphical assistance.
Note that while is defined for all of , is not, as the derivative of does not exist when . (From the left, the derivative approaches ; from the right the derivative is .) Thus one critical number of is .
So we have three important -values to consider: and . Evaluating at each gives, respectively, , and , shown in Figure 3.1.23. Thus the absolute minimum of is 1, the absolute maximum of is . Our answer is confirmed by the graph of in Figure 3.1.24.
The image shows the graph of a piecewise-defined function on the interval . For , the graph appears to be part of a parabola, descending from to . For , the graph is a straight line with positive slope, from to .
We have when and when . In general, when Thus when ( is always nonnegative so we ignore , etc.) So when . The only values to fall in the given interval of are and , where .
We again construct a table of important values in Figure 3.1.26. In this example we have five values to consider: . From the table it is clear that the maximum value of on is ; the minimum value is . The graph in Figure 3.1.27 confirms our results.
The graph of resembles a cosine wave, except that it is much flatter when is near . From an apparent relative maximum at , the graph descends in both directions toward relative minimum values close to the endpoints, at and .
A closed interval is not given, so we find the extreme values of on its domain. is defined whenever ; thus the domain of is . Evaluating at either endpoint returns 0.
The graph is the half of the unit circle , where . It has absolute minimum values at its endpoints, which are and . It reaches its (absolute and relative) maximum at the top of the circle, which is the point .
Using the The Chain Rule, we find . The critical points of are found when or when is undefined. It is straightforward to find that when , and is undefined when , the endpoints of the interval (which are in the domain of .) The table of important values is given in Figure 3.1.30. The maximum value is , and the minimum value is .
We have seen that continuous functions on closed intervals always have a maximum and minimum value, and we have also developed a technique to find these values. In Section 3.2, we further our study of the information we can glean from “nice” functions with the Mean Value Theorem. On a closed interval, we can find the average rate of change of a function (as we did at the beginning of Chapter 2). We will see that differentiable functions always have a point at which their instantaneous rate of change is same as the average rate of change. This is surprisingly useful, as we’ll see.