140 PRACTICAL MATHEMATICS 



Thus ~ is evidently a minimum at the point C, and if -~ 



is a minimum, then ^ = 0. 

 ax* 



Case II. When the angle a is obtuse. Let tan a = m. 

 Moving along the curve from A to C, the angle 6 is obtuse, and 

 increases to a. 



Then - is negative and increases to m. 



Moving along the curve from C to B, the angle 6 is obtuse, and 

 decreases from a. 



V 



Y 



CASE.I. 



X 0- 

 FIG. 43. 



CASEZ. 



X 



fir/ 



Then -p is negative and decreases from m. 



du du 



Thus - is evidently a maximum at the point C, and if -/ 



dx ax 



is a maximum, then -^ = 0. 



In general a point of inflexion may be denned as a point on a 

 curve at which the slope is greatest or least, while its position is 



given by the relation -^ = 0. 



Example. Find the points of inflexion of the curve 

 y = x* + 2X 3 - 36x 2 + 4>8x - 52 



dx 

 Hx* 



- 72x + 48 



-72 



