Local Maxima and Minima

So far in this chapter we have studied what are often referred to as global optimization problems. The reason for this terminology is that we have been seeking the absolutely largest or smallest values of a function, when we compare the function values at all points in the domain. In applied optimization problems, it is usually these global maxima and minima that are of interest. However, sometimes one is interested in the local maxima and minima of a function. In this case, we compare the function value at the point in question only with alternative function values at nearby points. For example, considering Fig. 9.8 and thinking of the graph as representing the profile of a landscape, mountain tops P\ and P2 represent local maxima, whereas valley bottoms Q\ and Q2 represent local minima.

FIGURE 9.8 Points <>| and cs are local maxima; ch and cfe are local minima.

FIGURE 9.8 Points <>| and cs are local maxima; ch and cfe are local minima.

If f(x) is defined on domain A, the precise definitions are as follows:

Function / has a local maximum at c if there is an interval (a, ß) about c such that f(x) < /(c) for all those x in A that also lie in (a, ß).

Function / has a local minimum at d if there is an interval (a. ß) about d such that f(x) > f(d) for all those x in A that also lie in (a, ß).

Note: These definitions imply that point a in Fig. 9.8 is a local minimum point and b is a local (and global) maximum point. Some authors restrict the definition of local maximum/minimum points only to interior points of the domain of the function. According to this definition, a global maximum that is not an interior point of the domain is not a local maximum point. We want a global maximum/minimum point always to be a local maximum/minimum point, so we stick to definitions [9.6] and [9.7],

It is obvious what we mean by local maximum/minimum values of a function, and the collective names are local extreme points and values.

In searching for maximum and minimum points, Theorem 7.4 of Section 7.2 is very useful. Actually, the same result is valid for local extreme points: At a local extreme point in the interior of the domain of a differentiable function, the derivative must be zero. This is clear if we recall that the proof of Theorem 7.4 was concerned only with the behavior of the function in a small interval about the optimal point. Consequently, in order to find possible local maxima and minima for a function / defined in an interval I, we can again search among the following types of points:

2. end points of I

3. points in / where f does not exist

We have thus established necessary conditions for a function / defined in an interval I to have a local extreme point. But how do we decide whether a point satisfying the necessary conditions is a local maximum, a local minimum, or neither? In contrast to global extreme points, it does not help to calculate the function value at the different points. To see why, consider again the function whose graph is given in Fig. 9.8. Point c\ is a local maximum point and d2 is a local minimum point, but the function value at c\ is smaller than the function value at d2.

The First-Derivative Test

There are two main ways of determining whether a given stationary point is a local maximum, a local minimum, or neither. One of them is based on studying the sign of the first derivative about the stationary point, and is an easy modification of [9.4] in Section 9.2.

Theorem 9.1 (The first-derivative test for local extrema)

(a) If f'(x) > 0 throughout some interval {a, c) to the left of c and f'(x) < 0 throughout some interval (c, b) to the right of c, then x = c is a local maximum point for f.

(b) If f'(x) < 0 throughout some interval (a, c) to the left of c and f'(x) > 0 throughout some interval (c, b) to the right of c, then x — c is a local minimum point for /.

(c) If f'(x) > 0 both throughout some interval (a. c) to the left of c and throughout some interval (c. b) to the right of c, then x = c is not a local extreme point for /. The same conclusion holds if f'ix) < 0 on both sides of c.

Only case (c) is not already covered by [9.4] in Section 9.2. In fact, if /'(x) > 0 in (a, c) and in (c, b), then f{x) is strictly increasing in (a, c] as well as in [c, b). Then x — c cannot be a local extreme point

Example 9.6

Classify the stationary points of /(x) = |x3 — ¿x2 — |x 4-1.

Solution In this case (see Example 9.4), we have f'(x) = | 1) (jc—2), so x = -1 and x = 2 are the stationary points. The sign diagram for f'(x) is:

We conclude from this sign diagram that x = — 1 is a local maximum point whereas x = 2 is a local minimum point.

Example 9.7

Classify the stationary points of

Solution Because x4 + x2 + 2 is >2 for all x, the denominator is never 0, so f{x) is defined for all x. Differentiation of f(x) yields fix) =

In order to study the sign Variation of f'(x), we must factorize x4 — x2 — 6. In fact, we have x4 - x2 - 6 = (x2)2 - ix2) - 6 = (x2 - 3)(x2 + 2) = (x -V3)(x + V3)(x2 + 2). Hence, fix) =

Both the denominator and the factor (x2 + 2) in the numerator are always positive. Hence, the sign variation of f'(x) is determined by the other factors in the numerator, as in the following sign diagram. Studying it we conclude from (a) in Theorem 9.1 that x = %/3 is a local maximum point, and from (b) that x = —\/3 is a local minimum point. According to (c), x = 0 is neither a local maximum, nor a local minimum point, because f'{x) > 0 in (-v/3,0) and in (0, V3).

------------<

-1

■ - - <

The graph is shown in Fig. 9.9. Note that /(-*) = - fix) for all x, so the graph of / is symmetric about the origin.

FIGURE 9.9

The Second-Derivative Test

For most problems of practical interest in which an explicit function is specified, Theorem 9.1 will determine whether a stationary point is a local maximum, a local minimum, or neither. Note that the theorem requires the knowledge of f'(x) at points in a neighborhood of the given stationary point. In the next sufficiency theorem, we need only properties of the function at the stationary point.

Theorem 92 (The Second-Derivative Test) Let / be a twice dif-ferentiable function in an interval I. Suppose c is an interior point of I. Then:

(a) /'(c) = 0 and /"(c) < 0 ==► c is a strict local maximum point.

(b) /'(c) = 0 and /"(c) > 0 ==> c is a strict local minimum point.

Proof To prove part (a), assume /'(c) = 0 and f"(c) < 0. By definition of /"(c) as the derivative of /'(x) at c.

Because /"(c) < 0, it follows from [*] that f'{c+h)/h < 0 if \h | is sufficiently small. In particular, if h is a small positive number, then f'(c + h) <0, so /' is negative in an interval to the right of c. In the same way, we see that /' is positive in some interval to the left of c. But then c is a strict local maximum point for /. Part (b) can be proved in the same way; for the inconclusive part (c), see the comments that follow.

Theorem 9.2 leaves unsettled case (c) when /'(c) = /"(c) = 0. Then "anything" can happen. Each of three functions /(x) = x4, f{x) = -x4, and f(x) = x3 satisfies /'(0) = /"(0) = 0. At x = 0, they have, respectively, a (local)

FIGURE 9.10

0 is a minimum point

0 is a maximum point

0 is an inflection point minimum, a (local) maximum, and a point of inflection, as shown in Figs. 9.10 to 9.12. Usually (as here), Theorem 9.1 can be used to classify stationary points at which f'(c) = /"(c) = 0. (For the definition of an inflection point, see [9.11] in Section 9.5.)

Theorem 9.2 can be used to obtain a useful necessary condition for local extrema. Suppose / is differentiable in the interval / and suppose that c is an interior point of I that is a local maximum point Then /'(c) = 0. Moreover, f"(c) > 0 is impossible, because by Theorem 9.2 (b) this inequality would imply that c is a strict local minimum. Hence, /"(c) has to be < 0. In the same way, we see that f"{c) > 0 is a necessary condition for local minimum. Briefly formulated:

The function studied in Example 9.7 is a typical example of when it is convenient to study the sign variation of the first derivative in order to classify the stationary points. (Using Theorem 9.2 requires finding f"(x), which is a rather involved expression.)

In theoretical economic models, it is more common to restrict the signs of second derivatives than to postulate a certain behavior in the sign variation of first derivatives. We consider a typical example.

Example 9.8

If a firm producing some'commodity has revenue function R(Q), cost function C(Q), and there is a sales tax of t dollars per unit, then Q* > 0 can only maximize profits provided that

(See Example 9.5 of Section 9.3, Equation [5].) Suppose R"(Q*) < 0 and C"(Q*) > 0. Equation [*] implicitly defines Q* as a differentiable function of t. Find an expression for dQ* jdt and discuss its sign. Also compute the derivative with respect to t of the optimal value rriQ*) of the profit function, and show that d7r(Q*)/dt = -Q*.

c is a local maximum for / =>• /"(c) < 0 c is a local minimum for / f"(c) > 0

Solution Differentiating [*] totally with respect to t yields dO* dO*

dt dt

Solving for dQ*/dt gives dQ* 1

The sign assumptions on R" and C" imply that dQ*/dt < 0. Thus, the optimal number of units produced will decline if the tax rate t increases.

The optimal value of the profit is it(Q*) = R(Q*) - C{Q*) - tQ*. Taking into account the dependence of Q* on r, we get dJt*(Q*) , dO* , dQ* dQ*

dt dt dt dt

where we used [*]. Thus, we see that by increasing the tax rate by one unit, the optimal profit will decline by Q* units. Note how the terms in dQ*/dt disappear from this last expression because of the first-order condition [*]. This is an instance of the "envelope theorem," which will be discussed in Section 18.7.

Example 9.9 (When to Harvest a Tree?)

Consider a tree that is planted at time t = 0, and let P(t) be its current market value at time t, where P(t) is differentiate. When should this tree be cut down in order to maximize its present discounted value? Assume that the interest rate is 100r% per year, compounded continuously.

Solution By using [8.28] in Section 8.5, the present value is fit) = Pit)e~n [1]

whose derivative is fit) = P'it)e~r' + Pit)i-r)e~n = e"" [P'(r) - rP(r)] [2]

A necessary condition for t* > 0 to maximize fit) is that fit*) = 0. We see from [2] that this occurs when

The tree, therefore, should be cut down precisely at time t* when the increase in the value of the tree over time interval (r*, t* + 1) (% P'it*)) is equal to the interest one would obtain over this time interval by investing amount Pit*) at interest rate r («s rP(t*)).

Let us look at the second-order condition. From [2], we find that

Evaluating f"(t) at t* and using [3] yields fit*) = e~rr [P"(t*) - rP'(t*j\ [4]

Assuming P(t*) > 0 and P"(t*) < 0, from [3] we have P'(t*) > 0. Then [4] gives fit*) < 0, so t* defined by [3] is a local maximum point. An example is given in Problem 4.

In this example, we did not consider how the ground the tree grows on may be used after cutting—for example, by planting a new tree. See Problem 5.

Note: In accepting maximization of present discounted value as a reasonable criterion for when a tree ought to be felled, one automatically dismisses the naive solution to the problem: Cut down the tree at the time when its current market value is greatest. Instead, the tree is typically cut down a bit sooner, because of "impatience" associated with discounting.

Problems

1. Consider the function / defined for all x by f(x)=x3- I2x

Find the two stationary points of / and classify them both by using the first-and second-derivative tests.

2. Determine all local extreme points and corresponding extreme values for the functions given by the following formulas:

3. A function f is given by the formula

a. Find the domain of /, the zeros of /, and the intervals where fix) is positive.

b. Find possible local extreme points and values.

c. Examine /(*) as x —► 0~, x —► 0+, and x oo. Also determine the limit of f'(x) as x —> oo. Has / a maximum or a minimum in the domain?

4. With reference to the tree-cutting problem of Example 9.9, consider the case where a. Find the value of r that maximizes f(t). Prove that the maximum point has been found.

5. Consider Example 9.9. Assume now that immediately after a tree is felled, a new tree of the same type is planted. If we assume that a new tree is planted at times r, 2r, 31, etc., then the present value of all the trees will be a. Find the sum of this infinite geometric series.

b. Prove that if f(t) has a maximum for some t* > 0, then and compare this condition to condition [3] in Example 9.9. 6. What requirements must be imposed on constants a, b, and c in order that a. will have a local minimum at x — 0?

7. Figure 9.13 graphs the derivative of a function /. Which of points a, b, c, d, and e are local maximum or minimum points for /?

8. Let function / be defined by

a. Find fix) and f"(*), and find the local extreme points of /.

b. Find the global extreme points, and draw the graph of /.

c. Use the previous results to find global extreme points for the function g defined for all x by g(x) = f(ex).

9. Consider the function fix) =

x4-x2 + l a. Compute fix) and find all local maximum and minimum points for /. Has / any global extreme points?

Harder Problems

10. Discuss local extreme points for the function fix) = x3 + ax + b. Use the result to show that the equation fix) = 0 has three different real roots if and only if 4a3 + 21b1 < 0.

11. Let / be defined for all x by fix) = (.x2 - 1)2/3.

b. Find the local extreme points of /, and draw the graph of /.

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