THE POINT OF INFLEXION 119 



CM/ 



always positive and -^ decreases to a zero value and then 

 increases. 



Since -^ decreases to a zero value and then increases as x in- 

 dx 



creases, the curve obtained by plotting x horizontally and -j- 



CUK 



vertically is of such a form that its lowest point occurs when 



dii 



- = ; hence at this point the tangent to the curve must be 



d 2 u 



horizontal, and so the slope or -=-4 is zero. 



dx z 



Thus at a point of inflexion on a curve we have the conditions 



(b) Moving along the curve from A to C, the angle 6 is obtuse 

 and increases to 180. 



Then -j- is negative and increases to 0. 



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

 decreases from 180. 



dii 

 Then -^ is negative and decreases from 0. 



00 



Therefore in the neighbourhood of a point of inflexion -j- is 

 always negative, and -^ increases to a zero value and then decreases. 



CL3C 

 fltJ 



Since -r- increases to a zero value and then decreases as x 

 ax 



increases, the curve obtained by plotting x horizontally and -j- 



vertically is of such a form that its highest point occurs when 



dti 



j- = ; hence at this point the tangent to the curve must be 



dhi . 

 horizontal, and so the slope or -^ is zero. 



010 



Thus at a point of inflexion on a curve we have the conditions 





