THE MINIMUM POINT 117 



If -j- is decreasing as x increases, then the curve obtained by 

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



-r? is negative. 



HB^ 



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



(2) -r^j is negative. 



72. Case II. The minimum point. Let C be the lowest point 

 of a curve, and at C - = (Fig. 38). 



FIG. 33. 



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 acute and 

 increases from 0. 



d\i 



Then -p is positive and increases from 0. 

 cue 



Therefore in the neighbourhood of a minimum point -^- is 



Iways increasing, and also -j- changes in sign from negative, 

 irough zero, to positive. 



