CONCAVITY AND CONVEXITY. 



315 



291. In the above figure P, Q, E, are points of inflexion 

 for the curves passing through them. At P there is a change 

 from concavity to convexity with respect to the axis of x. 

 At Q there is a point of inflexion, but the curve on both 

 sides of Q is convex to the axis of x. This agrees with 



d 2 y 

 Art. 289 ; since, if y and -& both change sign, no change 



(Ku 



occurs in the sign of their product. At R we have a point 



7 



~ 



72 



of inflexion at which ~ is infinite and therefore also ~. z 

 ctjc doc 



is infinite by Art. 113, a case which the investigation in 

 Art. 290 does not include. We should therefore in any 



d*y 

 example ascertain if -73, can become infinite, and if so we 



tZG 



must examine that case specially. We may trace the curve 

 in the neighbourhood of that point, or we may examine the 



d\ 

 sign of -~ for values of x differing slightly from that which 



gives rise to the infinite value, and thus determine if the curve 

 is concave or convex near the point in question. 



If we consider y as the independent variable, we may shew 

 in the manner of the preceding Articles, that a curve is convex 



d*x . 

 or concave to the axis of y, according as x -j 2 is positive or 



dfx ' 

 negative, and that at a point of inflexion -7-5 must vanish and 



change its sign. This is often useful in cases in which ~~ 

 becomes infinite. 



J2 . 



292. The connexion between -^~ and the concavity or 

 convexity of a curve, may also be shewn thus. 



