116 



PRACTICAL MATHEMATICS 



tan 6 will be negative and therefore -^ will be negative. Hence 



/y?y 

 when -J-, or the slope of a curve, is negative, y decreases as x 



increases. 



When we wish to find the highest or lowest point of a curve, 

 we do so by drawing the horizontal tangent to the curve and the 

 point of contact is the required point. As the tangent is horizontal, 



it is parallel to the axis of x and therefore its slope, or -j-, is zero. 



/77/ 



Thus the condition -2- = gives a point on the curve at which 



CUK 



the tangent is horizontal, and such a point may be either a maxi- 

 mum point, a minimum point, or a point of inflexion. 



e 



o 



FIG. 32. 



71. Case I. The maximum point. Let C be the highest point 

 dij 



of a curve, and at C 



dx 



= (Fig. 32). 



Moving along the curve from A to C, the angle 6 is acute and 



du 



decreases to 0. Then ~ is positive and decreases to 0. 

 doc 



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



dll 



decreases from 180. Then - is negative and decreases from 0. 



CtoC 



du 



Therefore in the neighbourhood of a maximum point ~- is 



dx 



always decreasing, and also ~ 

 through zero, to negative. 



changes in sign from positive, 



