120 PRACTICAL MATHEMATICS 



74. If the law of a curve is y =f(x), then will give the slope 

 of the curve at any point, the condition for a horizontal tangent 



fjft 



is obtained by putting -j- = 0. This is an equation to be solved 

 (tec 



for x, and the root will give a point on the curve at which the 

 tangent is horizontal. This point may be a maximum point, a 

 minimum point, or a point of inflexion. In order to decide which 



it happens to be, -=-^ is found and the value of x obtained by 



dii 



solving the equation ~ = is substituted in the resulting ex- 

 pression. If the result is negative, the point is a maximum point ; 

 if positive, a minimum point ; if zero, a point of inflexion. 



Example 1. State the nature of the points on the curve 

 y = 2x 3 9x 2 6Qx 25 at which the tangent is horizontal. 

 at . _ fysj QjT* _ fiOT 1 2^J 



-r- = 6x*- I8x - 60 

 dx 



The tangent is horizontal when ~ = 



That is, when Qx z - I8x - 60 = 



x* - 3x 10 = 



(x-5)(x+ 2) = 



x = 5 and x = 2 

 There are two points at which the tangent is horizontal. 



Now ~^ = I2x 18 



When x = 5, -7-^ = 42, a positive value, and y is a minimum 

 when x = 5. 



When x = 2, -7^- = 42, a negative value, and y is a maxi- 

 mum when x = 2. 



Then the expression 2x* 9x 2 60x 25 has its maximum 

 value 43 when x = 2, and its minimum value 300 when x = 5. 



Example 2. State the nature of the points on the curve 

 y = 3tf 4 Sx 3 24ex 2 + 96x 30 at which the tangent to the curve 

 is horizontal 



-30 



4>8x + 96 

 dx 



