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MAXIMA AND MINIMA
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See also:MAXIMA AND MINIMA , in See also: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 See also: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— I) can take all values from — 00 to — 1 and from + 7 to + , but has, so See also: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 See also:Pierre See also:Fermat. Particular cases had been discussed. Thus See also:Euclid in See also:book III. of the Elements finds the greatest and least straight lines that can be See also:drawn from a point to the circumference of a circle, and in book VI. (in a proposition generally omitted from See also:editions of his See also:works) finds the parallelogram of greatest See also:area with a given perimeter. See also:Apollonius investigated the greatest and least distances of a point from the perimeter of a conic See also:section, and discovered them to be the normals, and that their feet were the intersections of the conic with a rectangular See also:hyperbola. Some remarkable theorems on maximum areas are attributed to Zenodorus, and preserved by Pappus and See also:Theon of See also:Alexandria. The most noteworthy of them are the following: I. Of polygons of n sides with a given perimeter the See also:regular See also: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 See also:sphere is the See also:surface of given area which encloses the greatest See also:volume.
See also:Serenus of Antissa investigated the somewhat trifling problem of finding the triangle of greatest area whose sides are formed by the intersections with the See also:base and curved surface of a right circular See also:cone of a See also:plane drawn through its vertex.
The next problem on maxima and minima of which there appears to be any See also:record occurs in a See also:letter from See also:Regiomontanus to Roder (See also:July 4, 1471), and is a particular numerical example of the problem of finding the point on a given straight See also:line at which two given points subtend a maximum See also:angle. N. See also:Tartaglia in his General trattato de numeri et mesuri (c. 1556) gives, with-out See also:proof, a See also:rule for dividing a number into two parts such that the continued product of the See also: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 See also:differences of the values of a See also:function are insensible, a method virtually the same as that of the See also:differential calculus, and of See also:great use in dealing with geometrical maxima and minima. His method was See also:developed by See also:Huygens, See also:Leibnitz, See also:Newton and others, and in particular by See also: This is a useful method in the See also:case in which 4) (x) and f (x) are quadratics, but scarcely ever in any other case. The problem of finding the maximum product of n See also:positive quantities whose sum is given may also be found, algebraically, thus. If a and b are any two real unequal quantities whatever {2(a+b)}2>ab, so that we can increase the product leaving the sum unaltered by replacing any two terms by See also: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 See also: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' See also: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 See also: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 See also:interest or importance the reader may consult such a book as J. W. See also:Russell's A Sequel to Elementary See also:Geometry (See also:Oxford, 1907). The method of the differential calculus is theoretically very See also:simple. Let u be a function of several variables x1, x2, x3, ...x , supposed for the See also:present independent; if u is a maximum or minimum for the set of values x1, x2, x3, ... x,,, and u becomes u+Su, when xi, x2, x3, . . x„ receive 'small increments Soh, 8x2, . . . 6x,,; then Su must have the same sign for all possible values of 8x1, 8x2, . &e,,. 2 z Now Su=1bz1Sxi+ Ebzi20x12+2ESxIbx2Sx18x2 . . + . . The sign of this expression in general is that of E(Su/Sxi)Sxi, which cannot be one-signed when xi, x2, ... x„ can take all possible values, for a set of increments 8x1, 8x2, ... U. will give an opposite sign to the set—Sxi,—0x2, ... —Sx,,. Hence E(bu/oxi)Sxi must vanish for all sets of increments Sxi, . . . Sx„, and since these are independent, we must have bu/0x1=o, Su/0x2=o, Su/Sx„=o. A value of u given by a set of solutions of these equations is called a " See also:critical value " of u. The value of bu now becomes 3 Eaz 22 8x32+2 Z—; for u to be a maximum or minimum this must have always the samesign. For the case of a single variable x, corresponding to a value of x given by the equation du/dx=o, u is a maximum or minimum as d2u/dx2 is negative or positive. If d2u/dx2 vanishes, then there is no maximum or minimun unless d3u/dx3 vanishes, and there is a maximum or minimum according as d4u/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 See also:odd, there is no maximum or minimum. In the case of several variables, the quadratic z ESx 2bxi2+2 bxibxz 3xi3x2 + .. . must be one-signed. The See also:condition for this is that the See also:series of discriminants an , an an I all an a13 a21 a22 a21 a22 a23 an an an where apq denotes Stu/SapSaq 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 62u 82u / 022u )2 6x128x22 > 8xi6x2 for a maximum or minimum at all and S2u/8x12 and 62u/8x22 both negative for a maximum, and both positive for a minimum. It is important to See also:notice that by the quadratic being one-signed is meant that it cannot be made to vanish except when 8x1, 8x2, ... ox„ all vanish. If, in the case of two variables, S2u S2u 82., 2 8x12 8x22 = (Sxlax'') 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 See also:fourth See also:order must be taken See also:account of. Take for instance the function u x2—xy2+y3. Here the values x=o, y=o satisfy the equations Su/Sx=o, Su/Sy=o, 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 (8x)2, and are never negative. Here Su=8x2—SxOy2+0y3, and by putting 6x =o or an infinitesimal of the same order as 0y2, we can make the sign of Su depend on that of 0y3, and so be positive or negative as we please. On the other See also:hand, if we take the function u.=x2 —xy2+y4, x=o, y=o make zero a critical value of u, and here Su=0x2—SxOy2+Sy4, which is always positive, because we can write it as the sum of two squares, viz. (6x—4Sy2)2+4 6y4; 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 .'(x) =o has no roots in this See also:interval, it follows that o'(x) is one-signed as x increases from a to b, so that cti(x) is increasing or diminishing all the See also:time, and the greatest and least values of 0(x) are (ti(a) and ¢(b), though neither of them is a critical value. Consider the following example: A See also:person in a See also:boat a See also:miles from the nearest point of the See also:beach wishes to reach as quickly as possible a point b miles from that point along the See also:shore. The ratio of his See also:rate of walking to his rate of See also:rowing is cosec a. Where should he See also:land? Here let AB be the direction of the beach, A the nearest point to the boat 0, 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 B, so that OP=a sec 0, and PB=b—a tan B. If his rate of rowing is V miles an See also:hour his time will be a sec O/V+ (b—a tan B) See also:sin a/V See also:hours. See also:Call this T. Then to the first See also:power of 80, 0T= (a/V) sec2O (sin B—sin a)SB, so that if AOB>a, ST and SB have opposite signs from 0=o to 0=a, and the same signs from 0 = a to B = AOB. So that when AOB is > a, T decreases from B =0 to 0=a, and then increases, so that he should land at a point distant a tan a from A, unless a tan a>b. When this is the case, ST and 00 have opposite signs throughout the whole range of 0, so that T decreases as 0 increases, and he should See also:row See also: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, ... x,., not independent but connected by m functional relations ui=o, U 2 0 u,,,=o, 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 Sul =o, 6 1 1 2= o ,..., Sum =o. Consider instead of Su, what is the same thing, viz., Su+X1Su1+X25u2 + • • • + XmOum, where X2, . Xm, are arbitrary multipliers. The terms of the first order in this expression are Su Su,,, bxl 0x1 ex1 We can choose a1, ... A,,,, to make the coefficients of 6x1, 6x2, bxm, vanish, and the remaining bxm+1 to See also:axe may be regarded as independent, so that, when u has a critical value, their coefficients must also vanish. So that we put bx,+ X15 + ... + x"`bxm for all values of r. These equations with the equations u,=o, .. um =o are exactly enough to determine A,, . , x,,, 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 See also:illustration; consider the problem of deter-See also:mining the maximum and minimum radii vectors of the See also:ellipsoid x2,'See also:a2+y2lb2+z2/ c2 =1, where a2> b2> O. Here we require the maxi-mum and minimum values of x2 +y2 +z2 where x2la2+y2lb2+z21c2 = r. We have Su=2x0x (I ¢2) + 2yby (I + b2/ + 2111 (I + +Sx2 (1 + ~) + by2 (1 + )-O + bZ2 (I + -)c;). To make the terms of the first order disappear, we have the three equations : x(1 +X/a2) =o, y(1 +N/b2) =o, z(1 +X/c2) =o. These have three sets of solutions consistent with the conditions x2/a2+y2/b2+z2/c2=1, a2> b2> c2, viz.: (') y=o, z=o, A=—a2; (2) z=o, x=o, X= —b2; (3) x=o, y=o, A=—c2. In the case of (I) bu=0y2 (1—a2/b2) + Sze (I—a2/c2), which is always negative, so that u =a2 gives a maximum. In the case of (3) Su=0x2 (1—c2/a2)+Sy2 (1—c2/b2), which is always positive, so that u=c2 gives a minimum. In the case of (2) Su=Sx'(I-b2/a2)—Sz2(b2/c2—r), 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 See also:divide the ellipsoid into four portions in two of which a 2> r 2> b2, and in the other two b 2> r 2> O. (A. E. Additional information and CommentsThere are no comments yet for this article.
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