DIFFERENTIAL AND INTEGRAL CALCULUS. 79 



Hence the equation for ( -, ) is 8^ 2 + z - 1 = 0, therefore 

 VB/o 



1 + V(33) 



16 



6??/ 



This method should always be taken. The values of 



may be impossible ; thus if the equation be a*y*=x* (a; 2 -a 2 ), 

 the equation is satisfied by # = 0, j/ = 0, but we have 



77^ T* /?/\ 



^2 = - - t 1, and therefore the limit of (-) is V( 1). 



Such a point when the curve is traced should be conceived 

 as an infinitely small loop or, in this case, circle, the limit 

 of a finite one ; for if -the equation had been 

 a?y z = x(x b) (x* - a 2 ), b < a, 



there would be a loop of length 6, closing up to a point 

 when 5 = 0. 



MAXIMA AND MINIMA. 



If f(x) be any function of the independent variable x, 

 and we conceive x to increase uniformly from oo to + oo y 

 it will usually happen that/(x) is not always increasing 

 and not always decreasing, but that it sometimes does one 

 and sometimes the other. If a, 5, c be successive values of 

 a?, and if as x increases from - co to a, f(x) is always 

 increasing, but from a to 5, f(x) is always decreasing, 

 then /(a) is said to be a maximum value of f(x). If 

 from x = b to x = c, f(x) is again always increasing, 

 f(b) is said to be a minimum value of f(x] and so 

 on. That is, a maximum value is greater and a mini- 

 mum less than all adjacent values of /(#), although a maxi- 

 mum value need not be the greatest of all, nor a minimum 

 the least of all values of f(x). (Of course, however, this 

 may well be the case, and often is). Now if f(x] be 

 increasing as x increases, /' (a;) is positive ; if f(x] be 

 decreasing as x increases, /' (a;) is negative. Hence the 

 necessary and sufficient conditions for a maximum value of 



