118 



PRACTICAL MATHEMATICS 



If -Q- is increasing as x increases, then the curve obtained by 

 ax 



plotting x horizontally and -^ vertically is such that its slope or 



d 2 tf . 



-r4 is positive. 



dx 2 



Thus at a minimum point on a curve we have the conditions 



(2) -7^ is positive. 



73. Case III. The point of inflexion. Here we only have to 

 consider the case of the point of inflexion at which the tangent 

 to the curve is horizontal. 



O X O > 



FIG. 34. 



The horizontal line TjC^ touches the upper branch of the 

 curve at C and also the lower branch at C. 



ThenatC,^= (Fig. 34). 



(a) Moving along the curve from A to C, the angle is acute 

 and decreases to 0. 



Then -j- is positive and decreases to 0. 

 dx 



Moving along the curve from C to B the angle is acute and 

 increases from 0. 



Then -j- is positive and increases from 0. 



Therefore in the neighbourhood of a point of inflexion ^ is 



