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Topos Uranos
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Maxima and minima of a function

Summary

This lesson gathers the results of the preceding ones into a method for finding the extrema of a function. It calls the maximum and the minimum of a function on its domain absolute, to distinguish them from the local ones, and defines strict extrema. It then proves two tests that decide whether a point with zero derivative is a local extremum without studying the sign of the derivative around it: the second derivative test, which follows from the first derivative test, and the higher-order derivative test, which follows from L’Hôpital’s rule: if the first derivative that does not vanish at the point is of even order, there is an extremum, and if it is of odd order, there is none. The second half deals with absolute extrema. On a closed and bounded interval they exist, by the Weierstrass theorem, and Fermat’s theorem confines them to a list of candidates, the endpoints of the interval and the critical points, among which it suffices to compare values. On an open or unbounded interval they may fail to exist, and they are decided with monotonicity or with the limits at the endpoints of the interval; in particular, if a function continuous on an interval has a single critical point and attains a local extremum there, that extremum is absolute.

Learning objectives

  1. Distinguish absolute extrema from local ones, and strict extrema from those that are not, by means of quantifiers, and recognize each case in examples.
  2. Prove the second derivative test and the higher-order derivative test, and use them to decide whether a point with zero derivative is a local maximum, a local minimum or neither, even when the second derivative vanishes.
  3. Find the absolute extrema of a function continuous on a closed and bounded interval by comparing its values at the endpoints of the interval and at the critical points.
  4. Decide whether a function attains its absolute extrema on an open or unbounded interval, with monotonicity, with a single critical point or with the limits at the endpoints of the interval.

Absolute, local and strict extrema

The lesson on the Weierstrass theorem defined the maximum and the minimum of a function on a set, and the lesson on the theorems of Fermat, Rolle, Lagrange and Cauchy defined local extrema, which only compare the value at a point with the nearby values. It is now convenient to give the former a name that distinguishes them from the latter, and to specify when the value at the point is greater than all the others with which it is compared, and not merely greater than or equal to them.

DefinitionAbsolute extrema and strict extrema

Let ff be a function defined on a set AA and x0x_0 an element of AA. We say that ff attains at x0x_0 an absolute maximum if it attains at x0x_0 its maximum on AA, and an absolute minimum if it attains at x0x_0 its minimum on AA, in the sense of the lesson on the Weierstrass theorem; an absolute extremum is an absolute maximum or an absolute minimum. An absolute or local maximum is strict if f(x0)f(x_0) is greater than each of the other values with which it is compared:

(∀x∈A)(x≠x0→f(x)<f(x0)),(∃δ>0)(∀x∈A)(0<∣x−x0∣<δ→f(x)<f(x0))(\forall x \in A)\left(x \neq x_0 \rightarrow f(x) < f(x_0)\right), \qquad (\exists \delta > 0)(\forall x \in A)\left(0 < \abs{x - x_0} < \delta \rightarrow f(x) < f(x_0)\right)

The first formula defines the strict absolute maximum, and the second, the strict local maximum; strict minima are defined in the same way, with f(x)>f(x0)f(x) > f(x_0).

Absolute extrema are also called global, and local ones, relative. Every absolute extremum is local, with any δ\delta, and every strict extremum is an extremum; the converses are false. The function x3−3xx^3 - 3x attains at −1-1 a local maximum that is not absolute, as was seen in the lesson on the theorems of Fermat, Rolle, Lagrange and Cauchy. The cosine attains its absolute maximum on R\mathbb{R} at every multiple of 2π2\pi; none of these maxima is a strict absolute maximum, because the value 11 is repeated, and each one is a strict local maximum, because cos⁡x<1\cos x < 1 if 0<∣x−2kπ∣<2π0 < \abs{x - 2k\pi} < 2\pi. A constant function attains at every point an absolute maximum and an absolute minimum, and none is strict. In general, if ff attains at x0x_0 both a local maximum and a local minimum, it is constant near x0x_0. The absolute maximum value, if it exists, is unique; a strict absolute maximum, moreover, is attained at a single point.

Extrema depend on the domain. If ff attains an extremum at x0x_0 and BB is a subset of AA that contains x0x_0, the restriction of ff to BB also attains it, because it is compared with fewer values; the converse is false: x2x^2 attains on [1,2]\icc{1}{2} its absolute minimum at 11, and on R\mathbb{R} it has no extremum at 11.