DIFFERENTIAL AND INTEGRAL CALCULUS. 81 



of points for which the y is > 4, and similarly on the branch 

 beyond C an infinite number where y <0, i.e. is negative. 

 So in general if the curve y =f(x) be drawn, if y be a 

 maximum or minimum where/' (x) vanishes, the tangent 

 at such points is parallel to the axis of x. 



In (2) since ?/ 3 = (x af, we see that if y be negative, 

 x is impossible ; if y be positive x a has two equal and 



clii 

 opposite values, while at A, (OA = a) ~ is co ^ i.e. the 



tangent to the curve at A is perpendicular to OA. Hence 

 the curve is as (fig. 13), the point A being what is 

 called a cusp, and such a point always exists in the curve 

 yf(x], i--f(x) has a maximum or minimum value when 



/()-. 



As another example, take /(a?) or y (se-M) 4 (x - I) 6 ; 

 therefore 



Here/' (x) vanishes when x = 1, -J, and 1 ; also f (x) 

 is negative from -GO to - 1, positive from 1 to |, 

 negative from -J- to 1, and afterwards always positive. 

 Hence /( 1) or is a minimum value of /(a?), /( J), or 



8 4 x 1 2 6 



10 , or 1*223... is a maximum value, and /(I) or 



is again a minimum value. In this case we see at once 

 that y is always positive for real values of #, and, therefore, 

 that must be a minimum value. The form of the curve 

 (fig. 14) y f(x] in this case is somewhat as in the figure, 

 where 0^4 = 1, 6> = -l, 0<7=-, CD= 1-223... . 



In general those values of x which make f (x) = or 

 co should be selected, and it should be observed whether 

 the factor of/' (x) corresponding to each value has its index 

 odd or even (if that index be integral) ; if the index be odd 

 /' (x) must change sign as x passes through the corre- 

 sponding value, and there will be either a maximum or 

 minimum. But if the index be even (or of the form 



If 



