218 EXAMPLES OF MAXIMA AND MINIMA. 



55. Determine the cone of the greatest convex surface that 

 can be inscribed in a given sphere. 



Height =f r. 



5(5. Determine the cone so that its whole surface shall be 

 a maximum. 



Height = ^ (23 - 



57. Given the volume of a cylinder, find it's height and 



radius when the sum of the areas of its convex surface 

 and one end is a minimum. 



The height is equal to the radius. 



58. Of all cones described about a given sphere, find that of 



minimum volume. 



The sine of the semivertical angle must be J. 



59. A series of cones have their slant sides of the same 



length : find that which has the greatest volume. 



The tangent of the semivertical angle = \/2. 



60. Find the position of the chord which passes through a 



given point within a parabola, and cuts off from the 

 parabola the least possible area. 



61. Find a point in an ellipse from which, if perpendiculars 



be drawn to two given conjugate diameters, the sum 

 of their squares will be a maximum. 



62. Prove that <f> {/(#)} is necessarily either a maximum or 



minimum when f(x) is a maximum. And so also 

 when f (x) is a minimum. 



