MAXIMUM, in mathematics, denotes the greatest state or quantity attainable in a given case, or the greatest value of a variable quantity ; hence it stands oppos ed to the minimum, which is the least possible quantity in any case. Thus in the expression (4.-5 x, where a and b are constant, and x variable, the value of the expression will increase as b x or x dimi nishes, and it will be greatest, or a maxi mum, when x is least, or =0. The ex b pression a'— R: increases as niches, that is, as x increases, and it will be a maximum when x is infinite. If along the diameter, K Z (Plate X. Miscel. fig. 4.) of a circle, a perpendicular ordinate, L M, be conceived to move from K to Z, .it increases till it arrive at the centre, where it is greatest, and from thence it decreases till it vanishes at Z. Some quan tities continually increase, and have no maximum, unless what is infinite, as the ordinates of a parabola : some continual ly decrease, so that their minimum state is nothing, as the ordinates to the asymp totes of the hyperbola. Others increase to a certain point, which is their maxi mum, and then decrease again ; as the ordinates of a circle. Others admit of se veral maxima and minima ; as the ordi. nates of the curve (fig. 5 ) abcde, &c. where b and d are the maxima, and a c e are minima : hence it is easy to imagine of other variable quantities, exhibited by the ordinates of other kinds of curves. We have, under the article FLuxtosrs, given some examples on the maxima and mini ma of quantities : we shall in this place point out another mode of performing the same thing, with an example or two. The rule is this : " Find two values of an or dinate expressed in terms of the abscissa: put those two values equal to each other, striking out the parts that are common to both, and dividing all the remaining terms by the difference between the ab scissas, which will be a common factor in them : then supposing the abscissas to be come equal, that the equal ordinates may concur in the maximum or minimum, that difference will vanish, as well as all the terms of the equation that include it, and therefore, striking those terms out of the equation, the remaining terms will give the value of the abscissa corresponding to the maximum."
1. Suppose it were required to find the greatest ordinate in a semicircle K ill Q Z. Let KZ=a : K L the abscissa : L M the ordinate = y : hence L Z =a—x, and by the nature of the circle KLxLZ= L M., that is a Let the abscissa K X d, d being equal to L P ; the ordinate P Q=LM= y. KPxP Z=P Q., or x-I-dX a —x—d=ax—x.-2dx-Fad—d.
a — x. ; therefore — 2 d x a d—d.=0 : or ad = 2dx -Ed., or a= 2 x el, an equation derived from the equality of the two ordinates : now, by bringing the two equal ordinates toge ther, or making the two abscissas equal, their difference, d; vanishes, and a=2 x, or x =-. K N, the value of the ab scissa K N, when N 0 is a maximum, that is, the greatest ordinate bisects the dia meter.
2. Let it be required to divide a given line into two such parts, that the one drawn into the square of the other may be the greatest possible. Let the given line be a ; one part x, of course the other part a—x ; and therefbre by the terms of the question x' X a —x= a x. — xi is the product of one part by the square of the other. For the sake of comparison, let one part be x+d, then the other part will be a—x—d and .1.77C-al. X a—x--d • = a x.— x3 — 3 dx' + 2 a d — 3 d' X x a d'—di = (as before) a x' — X3 ; there fore, —3 d x. + 2 a d — 3 el; X a d' — d3, divided by d, gives — 3 x. 2 a-3 dXx+ a d— d., and now strik ing out the terms that have d in them, we get-3 x' + 2 a x =-- 0, and 3 x 2 a, and x = that is, the given line must be divided into two parts, in the ra tio of 3 to 2.