MAXIMA AND MINIMA 



121 



fflU 



The tangent is horizontal when -% = 0. 



That is, when 12^ - 24a: 2 - 48# + 96 - 



or x 3 - 2# 2 - 4a? + 8 = 



(z 2 -4)(,r-2) = 



<r = 2 and a; = - 2 

 There are two points at which the tangent is horizontal 



Now ^ = 8&r 2 - 48x - 48 



ax* 



<JrU 



When x = 2, -r-^ = 0, and a point of inflexion occurs when x = 2, 

 air 



d 2 // 

 When x = - 2, y^ = 192, a positive value, and t/ is a minimum 



CLJC 



when # = 2. 



Then the expression 3ar* &T 3 24# 2 + 96a? 30 has a minimum 

 value 20G when x = 2, and there is a point of inflexion when 

 x =2. 



75. Example 3. Find the dimensions of the cone of greatest 

 volume which can be cut from a sphere of given radius. 



FIG. 35. 



Let x be the perpendicular distance of the base of the cone 

 from the centre of the sphere and let R be the radius of the sphere. 

 Height of cone = R+ x 



Radius of base of cone = VR 2 x 2 



Volume of cone 





XT 



Now v is a maximum when -=- = 



ax 



