1911 Encyclopædia Britannica/Maxima and Minima
MAXIMA AND MINIMA, in mathematics. By the maximum or minimum value of an expression or quantity is meant primarily the “greatest” or “least” value that it can receive. In general, however, there are points at which its value ceases to increase and begins to decrease; its value at such a point is called a maximum. So there are points at which its value ceases to decrease and begins to increase; such a value is called a minimum. There may be several maxima or minima, and a minimum is not necessarily less than a maximum. For instance, the expression (x2 + x + 2)/(x − 1) can take all values from −∞ to −1 and from +7 to +∞, but has, so long as x is real, no value between -1 and +7. Here −1 is a maximum value, and +7 is a minimum value of the expression, though it can be made greater or less than any assignable quantity.
The first general method of investigating maxima and minima seems to have been published in A.D. 1629 by Pierre Fermat. Particular cases had been discussed. Thus Euclid in book III. of the Elements finds the greatest and least straight lines that can be drawn from a point to the circumference of a circle, and in book VI. (in a proposition generally omitted from editions of his works) finds the parallelogram of greatest area with a given perimeter. Apollonius investigated the greatest and least distances of a point from the perimeter of a conic section, and discovered them to be the normals, and that their feet were the intersections of the conic with a rectangular hyperbola. Some remarkable theorems on maximum areas are attributed to Zenodorus, and preserved by Pappus and Theon of Alexandria. The most noteworthy of them are the following:—
1. Of polygons of n sides with a given perimeter the regular polygon encloses the greatest area.
2. Of two regular polygons of the same perimeter, that with the greater number of sides encloses the greater area.
3. The circle encloses a greater area than any polygon of the same perimeter.
4. The sum of the areas of two isosceles triangles on given bases, the sum of whose perimeters is given, is greatest when the triangles are similar.
5. Of segments of a circle of given perimeter, the semicircle encloses the greatest area.
6. The sphere is the surface of given area which encloses the greatest volume.
Serenus of Antissa investigated the somewhat trifling problem of finding the triangle of greatest area whose sides are formed by the intersections with the base and curved surface of a right circular cone of a plane drawn through its vertex.
The next problem on maxima and minima of which there appears to be any record occurs in a letter from Regiomontanus to Roder (July 4, 1471), and is a particular numerical example of the problem of finding the point on a given straight line at which two given points subtend a maximum angle. N. Tartaglia in his General trattato de numeri et mesuri (c. 1556) gives, without proof, a rule for dividing a number into two parts such that the continued product of the numbers and their difference is a maximum.
Fermat investigated maxima and minima by means of the principle that in the neighbourhood of a maximum or minimum the differences of the values of a function are insensible, a method virtually the same as that of the differential calculus, and of great use in dealing with geometrical maxima and minima. His method was developed by Huygens, Leibnitz, Newton and others, and in particular by John Hudde, who investigated maxima and minima of functions of more than one independent variable, and made some attempt to discriminate between maxima and minima, a question first definitely settled, so far as one variable is concerned, by Colin Maclaurin in his Treatise on Fluxions (1742). The method of the differential calculus was perfected by Euler and Lagrange.
John Bernoulli’s famous problem of the “brachistochrone,” or curve of quickest descent from one point to another under the action of gravity, proposed in 1696, gave rise to a new kind of maximum and minimum problem in which we have to find a curve and not points on a given curve. From these problems arose the “Calculus of Variations.” (See Variations, Calculus of.)
The only general methods of attacking problems on maxima and minima are those of the differential calculus or, in geometrical problems, what is practically Fermat’s method. Some problems may be solved by algebra; thus if y = ƒ(x) ÷ φ(x), where ƒ(x) and φ(x) are polynomials in x, the limits to the values of yφ may be found from the consideration that the equation yφ(x) − ƒ(x) = 0 must have real roots. This is a useful method in the case in which φ(x) and ƒ(x) are quadratics, but scarcely ever in any other case. The problem of finding the maximum product of n positive quantities whose sum is given may also be found, algebraically, thus. If a and b are any two real unequal quantities whatever 12(a + b)}2 > ab, so that we can increase the product leaving the sum unaltered by replacing any two terms by half their sum, and so long as any two of the quantities are unequal we can increase the product. Now, the quantities being all positive, the product cannot be increased without limit and must somewhere attain a maximum, and no other form of the product than that in which they are all equal can be the maximum, so that the product is a maximum when they are all equal. Its minimum value is obviously zero. If the restriction that all the quantities shall be positive is removed, the product can be made equal to any quantity, positive or negative. So other theorems of algebra, which are stated as theorems on inequalities, may be regarded as algebraic solutions of problems on maxima and minima.
For purely geometrical questions the only general method available is practically that employed by Fermat. If a quantity depends on the position of some point P on a curve, and if its value is equal at two neighbouring points P and P′, then at some position between P and P′ it attains a maximum or minimum, and this position may be found by making P and P′ approach each other indefinitely. Take for instance the problem of Regiomontanus “to find a point on a given straight line which subtends a maximum angle at two given points A and B.” Let P and P′ be two near points on the given straight line such that the angles APB and AP′B are equal. Then ABPP′ lie on a circle. By making P and P′ approach each other we see that for a maximum or minimum value of the angle APB, P is a point in which a circle drawn through AB touches the given straight line. There are two such points, and unless the given straight line is at right angles to AB the two angles obtained are not the same. It is easily seen that both angles are maxima, one for points on the given straight line on one side of its intersection with AB, the other for points on the other side. For further examples of this method together with most other geometrical problems on maxima and minima of any interest or importance the reader may consult such a book as J. W. Russell’s A Sequel to Elementary Geometry (Oxford, 1907).
The method of the differential calculus is theoretically very simple. Let u be a function of several variables x1, x2, x3 . . . xn, supposed for the present independent; if u is a maximum or minimum for the set of values x1, x2, x3, . . . xn, and u becomes u + δu, when x1, x2, x3 . . . xn receive small increments δx1, δx2, . . . δxn; then δu must have the same sign for all possible values of δx1, δ2 . . . δxn.
Now
| δu = Σ | δu | δx1 + 12 { Σ | δ2u | δx12 + 2Σ | δ2u | δx1δx2 . . . } + . . . . |
| δx1 | δx12 | δx1δx2 |
The sign of this expression in general is that of Σ(δu/δx1)δx1, which cannot be one-signed when x1, x2, . . . xn can take all possible values, for a set of increments δx1, δx2 . . . δxn, will give an opposite sign to the set −δx1, −δx2, . . . −δxn. Hence Σ(δu/δx1)δx1 must vanish for all sets of increments δx1, . . . δxn, and since these are independent, we must have δu/δx1 = 0, δu/δx2 = 0, . . . δu/δxn = 0. A value of u given by a set of solutions of these equations is called a “critical value” of u. The value of δu now becomes
| 12 { Σ | δ2u | δx12 + 2 Σ | δ2u | δx1δx2 + . . . }; |
| δx12 | δx1δx2 |
for u to be a maximum or minimum this must have always the same sign. For the case of a single variable x, corresponding to a value of x given by the equation du/dx = 0, u is a maximum or minimum as d 2u/dx2 is negative or positive. If d 2u/dx2 vanishes, then there is no maximum or minimum unless d 2u/dx2 vanishes, and there is a maximum or minimum according as d 4u/dx4 is negative or positive. Generally, if the first differential coefficient which does not vanish is even, there is a maximum or minimum according as this is negative or positive. If it is odd, there is no maximum or minimum.
In the case of several variables, the quadratic
| Σ | δ2u | δx12 + 2 Σ | δ2u | δx1δx2 + . . . |
| δx12 | δx1δx2 |
must be one-signed. The condition for this is that the series of discriminants
| a11 , | a11 a12 | , | a11 a12 a13 | , . . . |
| a21 a22 | a21 a22 a23 | |||
| a31 a32 a33 |
where apq denotes δ2u/δapδaq should be all positive, if the quadratic is always positive, and alternately negative and positive, if the quadratic is always negative. If the first condition is satisfied the critical value is a minimum, if the second it is a maximum. For the case of two variables the conditions are
| δ2u | · | δ2u | > ( | δ2u | )2 |
| δx12 | δx22 | δx1δx2 |
for a maximum or minimum at all and δ2u/δx12 and δ2u/δx22 both negative for a maximum, and both positive for a minimum. It is important to notice that by the quadratic being one-signed is meant that it cannot be made to vanish except when δx1, δx2, . . . δxn all vanish. If, in the case of two variables,
| δ2u | · | δ2u | = ( | δ2u | )2 |
| δx12 | δx22 | δx1δx2 |
then the quadratic is one-signed unless it vanishes, but the value of u is not necessarily a maximum or minimum, and the terms of the third and possibly fourth order must be taken account of.
Take for instance the function u = x2 − xy2 + y2. Here the values x = 0, y = 0 satisfy the equations δu/δx = 0, δu/δy = 0, so that zero is a critical value of u, but it is neither a maximum nor a minimum although the terms of the second order are (δx)2, and are never negative. Here δu = δx2 − δxδy2 + δy2, and by putting δx = 0 or an infinitesimal of the same order as δy2, we can make the sign of δu depend on that of δy2, and so be positive or negative as we please. On the other hand, if we take the function u = x2 − xy2 + y4, x = 0, y = 0 make zero a critical value of u, and here δu = δx2 − δxδy2 + δy4, which is always positive, because we can write it as the sum of two squares, viz. (δx − 12δy2)2 + 34δy4; so that in this case zero is a minimum value of u.
A critical value usually gives a maximum or minimum in the case of a function of one variable, and often in the case of several independent variables, but all maxima and minima, particularly absolutely greatest and least values, are not necessarily critical values. If, for example, x is restricted to lie between the values a and b and φ′(x) = 0 has no roots in this interval, it follows that φ′(x) is one-signed as x increases from a to b, so that φ(x) is increasing or diminishing all the time, and the greatest and least values of φ(x) are φ(a) and φ(b), though neither of them is a critical value. Consider the following example: A person in a boat a miles from the nearest point of the beach wishes to reach as quickly as possible a point b miles from that point along the shore. The ratio of his rate of walking to his rate of rowing is cosec α. Where should he land?
Here let AB be the direction of the beach, A the nearest point to the boat O, and B the point he wishes to reach. Clearly he must land, if at all, between A and B. Suppose he lands at P. Let the angle AOP be θ, so that OP = a secθ, and PB = b − a tan θ. If his rate of rowing is V miles an hour his time will be a sec θ/V + (b − a tan θ) sin α/V hours. Call this T. Then to the first power of δθ, δT = (a/V) sec2θ (sin θ − sin α)δθ, so that if AOB > α, δT and δθ have opposite signs from θ = 0 to θ = α, and the same signs from θ = α to θ = AOB. So that when AOB is > α, T decreases from θ = 0 to θ = α, and then increases, so that he should land at a point distant a tan α from A, unless a tan α > b. When this is the case, δT and δθ have opposite signs throughout the whole range of θ, so that T decreases as θ increases, and he should row direct to B. In the first case the minimum value of T is also a critical value; in the second case it is not.
The greatest and least values of the bending moments of loaded rods are often at the extremities of the divisions of the rods and not at points given by critical values.
In the case of a function of several variables, X1, x2, . . . xn, not independent but connected by m functional relations u1 = 0, u2 = 0, . . ., um = 0, we might proceed to eliminate m of the variables; but Lagrange’s “Method of undetermined Multipliers” is more elegant and generally more useful.
We have δu1 = 0, δu2 = 0, . . ., δum = 0. Consider instead of δu, what is the same thing, viz., δu + λ1δu1 + λ2δu2 + . . . + λm δum, where λ1, λ2, . . . λm, are arbitrary multipliers. The terms of the first order in this expression are
| Σ | δu | δx1 + λ1 Σ | δu1 | δx1 + . . . + λm Σ | δum | δx1. |
| δx1 | δx1 | δx1 |
We can choose λ1, . . . λm, to make the coefficients of δx1, δx2, ... δxm, vanish, and the remaining δxm+1 to δxn may be regarded as independent, so that, when u has a critical value, their coefficients must also vanish. So that we put
| δu | + | δu1 | + . . . + λm | δum | = 0 |
| δxr | δxr | δxr |
for all values of r. These equations with the equations u1 = 0, . . ., um = 0 are exactly enough to determine λ1, . . ., λm, x1 x2, . . ., xn, so that we find critical values of u, and examine the terms of the second order to decide whether we obtain a maximum or minimum.
To take a very simple illustration; consider the problem of determining the maximum and minimum radii vectors of the ellipsoid x2/a2 + y2/b2 + z2/c2 = 1, where a2 > b2 > c2. Here we require the maximum and minimum values of x2 + y2 + z2 where x2/a2 + y2/b2 + z2/c2 = 1.
We have
| δu = 2xδx ( 1 + | λ | ) + 2yδy ( | λ | ) + 2zδz ( | λ | ) |
| a2 | b2 | c2 |
| + δx2 ( 1 + | λ | ) + δy2 ( | λ | ) + δz2 ( | λ | ). |
| a2 | b2 | c2 |
To make the terms of the first order disappear, we have the three equations:—
x2/a2 + y2/b2 + z2/c2 = 1, a2 > b2 > c2, viz.:—
| (1) y = 0, z = 0, λ = −a2; (2) z = 0, x = 0, λ = −b2; |
| (3) x = 0, y = 0, λ = −c2. |
In the case of (1) δu = δy2 (1 − a2/b2) + δz2 (1 − a2/c2), which is always negative, so that u = a2 gives a maximum.
In the case of (3) δu = δx2 (1 − c2/a2) + δy2 (1 − c2/b2), which is always positive, so that u = c2 gives a minimum.
In the case of (2) δu = δx2 (1 − b2/a2) − δz2(b2/c2 − 1), which can be made either positive or negative, or even zero if we move in the planes x2 (1 − b2/a2) = z2 (b2/c2 − 1), which are well known to be the central planes of circular section. So that u = b2, though a critical value, is neither a maximum nor minimum, and the central planes of circular section divide the ellipsoid into four portions in two of which a2 > r 2 > b2, and in the other two b2 > r 2 > c2. (A. E. J.)