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 {1⁄2(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 lo 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.