The Extreme Values, or Extrema

Extreme Values The extreme values, or extrema, of a function are places where a function reaches its minimum and maximum values. Extreme values come into use in both design scenarios, as well as visualizing functions. Suppose we are designing the packaging for a product. In such a situation it would be beneficial t o m inimize t he w aste u sed i n p roduction, i n o rder t o c onserve r esources. I f w e h ave a function that describes the amount of waste used in production, it would be our goal to find its minimum, and use that to guide our production process. These types of design problems are called optimization problems, where we either want to maximize a benefit, o r m inimize a l oss. Solving an optimization problem comes down to finding t he e xtreme v alues o f a f unction m odeling the situation, and choosing a suitable operating point. We will also see extreme values play a vital role in sketching functions, where we can use a very limited amount of information about a function in order to gain a lot of information about its behavior. In order to graph, and reason visually about functions, we will need to: identify the function’s extrema, find i ntervals o ver w hich t he f unction i s i ncreasing a nd d ecreasing, and determine the concavity of the function. We can accomplish the first o f t hese two t asks u sing the first d erivative, a nd t he l ast u sing t he s econd d erivative. We w ill b egin w ith fi nding th e extrema, or extreme values of a given function. We can identify two types of extrema - local and global. Global extrema are the largest and smallest values that a function takes on over its entire domain, and local extrema are extrema which occur in a specific neighborhood of the f unction. In b oth the local and global cases, it is important to be cognizant of the domain over which the function is defined. T hat w hich i s a n e xtremum on one domain may very well not be over a new domain, and vice versa. Before delving further, let us give the formal definitions o f t hese various extrema. Definition 0 .1.1 ( Global E xtrema). T he f unction f : D → R h as a g lobal m aximum a t c ∈ D if f (x) ≤ f (c) for all x ∈ D. We say that f has a global minimum at c ∈ D if f (x) ≥ f (c) for all x ∈ D. Definition 0 .1.2 ( Local E xtrema). T he f unction f : D → R h as a l ocal m aximum a t c ∈ D if f (x) ≤ f (c) for all x in some open interval centered at c. We say that f has a local minimum at c ∈ D if f (x) ≥ f (c) for all x in some open interval centered at c. If c is an interior point of D we require that the interval extend on both sides of c; if c is at an endpoint of the domain, we only require that the above inequalities hold on the portion of the interval that is in the domain D. Note that in the above definitions f or l ocal e xtrema, t he i nterval w e fi nd ma y be qu ite small. Nevertheless, if we can find a s ingle i nterval over w hich t he a bove i nequalities h old, t hen t hey will also hold for any smaller interval. Since any interval of finite l ength c ontains a n i nfinite number of points, there is no way we can test every point to look for extrema. Instead, we will need to be more clever with the places we look for extrema. We need to emphasize that we are only interested in identifying the extrema of continuous functions. If we are faced with a discontinuous function, 1 then it may very well jump to an exceedingly high or low value at any strange point, which would make tracking down extrema a very difficult endeavor. As stated earlier, the first derivative is the tool we will need to use to find global and local extrema. Recall that the first derivative tells us whether a function is increasing, decreasing or neither, and if it is changing, how quickly. Here we are not so interested in how quickly the function is changing, but whether or not it is changing, and in which direction; we gain this information from the sign of the first derivative: if the derivative is positive, the function is increasing, if it is negative, the function is decreasing, and if it is zero the function is not changing. If the derivative has the same sign over an entire interval, then we can say that the function is either increasing, decreasing, or not changing over that interval. Formally, for a function to be increasing on an interval means that (x1 < x2 =⇒ f (x1 ) < f (x2 )), and for the function to be decreasing (x1 < x2 =⇒ f (x1 ) > f (x2 )). A careful observer may already be able to deduce how we can use the first derivative to find extrema. If we want to identify a local maximum at a point c, we want the function to increase to the left of c, and to decrease to the right of c. This will put c at a peak in the function. If the function is increasing to the left of c, the derivative must be positive to the left of c, and if the function is decreasing to the right of c, the derivative must be negative to the right of c. When we say that the function is increasing to the left of c, we mean there is some interval (b, c) so that the derivative is positive on (b, c), and when we say the function is decreasing to the right of c, we mean there is some interval (c, d) so that the derivative is negative on (c, d). It is only possible for a derivative to go from positive to negative in one of two ways: either the derivative must cross 0 (which it would if the derivative is a continuous function), or the derivative must be discontinuous, so that it can jump from positive to negative (for instance in the corner of |x|/x at x = 0). The point at which the derivative should cross zero (or fail to exist) is c, the local maximum. Similar analysis would indicate that the derivative should be zero (or not exist) at a local minimum, but with the derivative negative to the left, and positive to the right. In conclusion, we can only have a local extremum at a point c if the sign of the derivative changes around c (either from positive to negative - a maximum, or from negative to positive - a minimum). And the only way the derivative can change signs around c is if f 0 (c) = 0 or f 0 (c) does not exist. Such points are called critical points. Definition 0.1.3 (Critical Point). An interior point of the domain of a function f where f 0 is either zero or undefined is a critical point of f . Apropos to the previous discussion, the only place in the interior of a function’s domain that it can have a local extremum is at a critical point (but a critical point doesn’t guarantee there is a local extremum). The only points left to consider are the endpoints of the domain. Thus, if we want to find extrema, we need to look at critical points, and the endpoints of the domain. Now we have transformed the problem of evaluating the function at an infinite number of points, to evaluating it at a few, select points - much easier. Once we have found the local extrema, it is an easy task to find the global extrema. If a function has a global maximum on a given domain, it will correspond to the local maximum with the largest value. Similarly, a global minimum corresponds to the local minimum with the smallest value. Just as we would expect from the previous discussion, if a function f has an extremum at an interior point c of its domain, and f 0 (c) is defined, then f 0 (c) = 0. However, the converse of this statement is untrue - the fact that f 0 (c) = 0 does not guarantee that c is a local extremum. For instance, we could have a function where the derivative is positive, becomes zero at a point c, and then becomes positive again (x3 is an example of this, at x = 0). Since the function is increasing to the right of c, it is clear that c cannot be a local maximum, and since the function is increasing to 2 the left of c, c also cannot be a local minimum. For any function that isn’t a constant, we only have a local extremum at c when the derivative changes signs around c; if the derivative doesn’t change signs, then we do not have a local extremum. In conclusion, a zero derivative doesn’t guarantee a local extremum, but it is a very good place to try and identify one. We may also find extrema at points where the derivative is undefined (because the derivative could jump and change signs at such a discontinuity). It turns out that a continuous function defined on a closed interval always achieves its global maximum and minimum. Theorem 0.1.1 (Extreme Value Theorem). Let f : [a, b] → R be continuous. It follows that f attains both a global minimum value m and a global maximum value M in the interval [a, b]. That is, there exist x1 , x2 ∈ [a, b] so that f (x1 ) = m and f (x2 ) = M , and m ≤ f (x) ≤ M for all x ∈ [a, b]. In contrast, it is possible for a function to not have any extreme values on a given domain. Consider the function f : [0, 1) → R, f (x) = x. The smallest value is f (0) = 0, which corresponds to the global minimum. However, there is no global maximum, as no matter how close a point is to x = 1, one can choose a point even closer that yields a larger function value. If the function was defined on 0 < x ≤ 1, then it would instead have a global maximum but no global minimum. If we defined this function on (0, 1), then it would have no extreme values. In contrast, if we define the function on the closed interval [0, 1] then it has both a global maximum and minimum, as required by the extreme value theorem. Example 1. Find all minima and maxima of f (x) = xe−x on the domain [0, 4]. Solution To find local extrema, we need to look at the behavior of the first derivative around critical points, as well as at endpoints. We find f 0 (x) = 1 · e−x + x · (−e−x ) = (1 − x)e−x . Solving for f 0 (x) = 0 we find 0 = (1 − x)e−x . The right-hand side is 0 only if one of the two terms is zero, and since e−x is always nonnegative, we find that x = 1 is the only critical point. According to our previous discussion, we know that the derivative of a function can only change signs around a critical point, so if we partition the interval [0, 4] into subintervals, we know that the sign of the derivative must be constant on [0, 1), as well as on (1, 4]. If the signs of the derivatives differ on these two subintervals we will know that we have found an extremum. For x < 1 we have 1 − x > 0, and for x > 1 we have 1 − x < 0, so this critical point is a local maximum. On the endpoints we find f 0 (0) = 1 > 0, so the left endpoint is a local minimum (as the function is increasing to the right of it). Similarly, we find f 0 (4) = −3e−4 ≈ −0.055 < 0, so the right endpoint is also local minimum (as the function is decreasing to the left of it). To find the global minima and maxima, we need to compare the function values at all critical points, as well as the endpoints of the domain. We find that f (0) = 0, f (1) = e−1 ≈ 0.367, and f (4) = 4e−4 ≈ 0.073. Thus, f (0) is both a local and global minimum, and f (1) is both a local and global maximum. It is important to once again note that the global and local maxima are domain specific - that is, the points at which they occur may be different for different domains of the function (which they certainly are in this case). The graph of this function and its derivative are given in figure 1. Example 2. Find the global minima and maxima of f (x) = x3 − x2 on the interval [−1 : 1]. Solution The first step is to identify the critical points. To do so, we compute f 0 (x) = 3x2 − 2x. 3 0.4 1 0.75 (1 − x)e−x xe−x 0.3 0.2 0.5 0.25 0.1 0 0 −0.25 0 1 2 x 3 4 0 1 (a) 2 x 3 4 (b) Figure 1: xe−x and its derivative. Setting the derivative equal to 0 we find 0 = 3x2 −2x = x(3x−2), so we have critical points at x = 0 and x = 32 . Comparing the values of the critical points and endpoints, we find that: f (−1) = −2, f (0) = 0, f ( 23 ) ≈ −0.15, and f (1) = 0. Thus, the global minimum is f (−1), while both f (0) and f (1) are global maxima. 0 5 −0.5 0 x3 − x2 x3 − x2 Example 3. Find the global minima and maxima of f (x) = x3 − x2 on the interval [−2 : 2]. Solution We have already found the critical points, so we simply need to compare their values to those of the new endpoints. f (−2) = −12 and f (2) = 4, so clearly f (−2) is the global minimum, and f (2) is the global maximum. Note that on the new domain there is a single global maximum, rather than two (see figure 2). −1 −5 −10 −1.5 −15 −2 −1 0 x −0.5 0.5 1 −2 (a) 0 x −1 1 2 (b) Figure 2: The extreme values of x3 − x2 depend on the domain. Example 4. Find the global maxima and minima of f (x) = |x| over the domain −2 ≤ x ≤ 3. Solution In order to differentiate this function, it is useful to define it piecewise as follows  x x≥0 f (x) = −x x < 0 We can see that for x > 0 we have f 0 (x) = 1, and for x < 0 we have f 0 (x) = −1, so we don’t obtain any critical points there. However, the derivative is not defined at x = 0 (because it is a corner), so we have a critical point at x = 0 (and it turns out to be a local minimum, because the derivative 4 changes signs from negative to positive as we cross it). The values of the endpoints and this critical point are f (−2) = 2, f (0) = 0, and f (3) = 3. Thus, x = 0 is the global minimum, and x = 3 is the global maximum. 5