WHEN THE DIFFERENTIAL COEFFICIENT IS INFINITE. 207 



< (a) in the former case is a minimum value of <f> (x} and 

 in the latter a maximum value. 



Also, since a is a proper fraction, 



^ \, ' is infinite when h = 0, 



an 



therefore <J> (x) is infinite when x = a. 



Hence < (x) may be a maximum or a minimum when <j> (a-) 

 is infinite. 



Example. Suppose 



$ (x) = c + (a; - a ) ? + (x - a) * ; 



therefore </> (a + h) = c + h l + h* , 



Hence </> (a + K) and <j> (a fi) are both necessarily greater 

 than </> (a). Hence <j> (a) is a minimum value of </> (x), and 

 it is obvious that <f> (x} is infinite when x = a. 



222. Ow certain cases of Geometrical Maxima and Minima. 



We occasionally meet in Geometry cases of maxima or 

 minima values for which the ordinary analytical process 

 appears to fail, though from geometrical considerations it is 

 obvious that maxima or minima do exist. The following 

 problem will introduce the difficulty which it is proposed to 

 explain. " Find the maximum and minimum perpendicular 

 from the focus on the tangent to an ellipse, the perpendicular 

 being expressed in terms of the radius vector." 



The equation which gives the perpendicular in terms of 

 the radius vector is 



2 _ R'r 

 P ~2a-r ) 



therefore p~- =777^ ^ > which must = 0. 



* dr (2a - ry 



Now this can only be satisfied by r = oo , which values 

 are not admissible, whereas we .know from Geometry that p 



