316 POINTS OF INFLEXION. 



Let PL, QM, RN, be three equidistant ordinates. Draw 

 the chord PR meeting QM at H. 

 Let y ^(x) be the equation to the ff Q 



curve; x, y, the co-ordinates of P; -r^fa 



LM= MN= h. If the curve be con- 

 cave to the axis of x, QM is greater 

 than HM; and therefore 2QM 

 greater than 2NM, that is, greater d JG JH 

 Hence 



<f> (x + 2K) 2<f> (as + K) + <f> (x) is negative, 



. , d> (x + 2&) - 2< (x + h} + <b (x) . 

 and therefore also J-S 1 TJL_ L 1S negative. 



Let 7^ diminish indefinitely, and it follows by Art. 127, 

 that </>" (x) is negative. Similarly, if the curve be convex 

 to the axis of x, then <j)" (x) is positive. 



293. We will briefly indicate another method by which 

 the results of this Chapter are sometimes ob.tained. It is either 

 deduced from some definition of concavity and convexity, or 

 given as the definition of those words, that y being supposed 



positive, a curve is convex to the axis of #, if -j- be increasing, 



CtdC 



that is, if -73 be positive, and concave if -p be decreasing, that 



*<*Yu 



is, it -j be negative. 



Also a point of inflexion may be defined as a point where 

 the curve changes from being concave to being convex, or 



d a y 



vice versa. Hence -r^ must change sign at a point of inflexion. 

 ax 



A point of inflexion may also be defined as a point at 

 which the inclination of the tangent to the axis has a maxi- 

 mum or minimum value. Since when this angle has a maxi- 

 mum or minimum value, so also has its tangent, we must 



have -^ a maximum or minimum at a point of inflexion. 

 Hence -y^ must change sign. 



