CONCAVITY AND CONVEXITY. 313 



still be taken as a consequence of the meaning of the terms 

 concave and convex. 



Let y^ denote the ordinate of the curve corresponding 

 to the abscissa x + h, and y z the corresponding ordinate to 

 the tangent at P. If # = </> (x) be the equation to the curve, 

 we have 



yi =0 (x) + Af (x) + ~ 4>"(as + 0h). 



A 



And since the equation to the tangent at P is 



Y-y = 4>'(x)(X-x), 

 we have 



y* = OB) + ^$' 0) ; 



A 2 

 therefore ^ y 2 = <" (x + 0A) . 



ZJ 



This, if we take A small enough, will have the same sign 

 as</>"(:r); and therefore the curve is convex to the axis of 

 x if (f)" (x) be positive, and concave if <" (x) be negative. 



We have supposed in the figures that the curve is above the 

 axis of x. If it be Mow the axis of x, then y^ and y z are 

 the numerical values of the ordinates, and the curve is convex 

 if y l + y 2 be positive, that is, if <j>"(x) be negative, and con- 

 cave if <$' (x) be positive. 



Both cases, may be included in one enunciation, thus, "A 



curve is convex or concave to the axis of x according as y ~, 



J ax 



is positive or negative." 



290. DEFINITION. A point of inflexion is a point at 

 which a curve cuts its tangent at that point. 



To find the conditions for the existence of a point of 

 inflexion. Let y = (f>(x] be the equation to a curve ; let 

 x, y, be the co-ordinates of a point in a curve, and x + h, y v 

 the co-ordinates of an adjacent point. Let the tangent of 

 the curve at the point (x, y) be drawn, and let y z be the 



