EXAMPLES OF MAXIMA AND MINIMA. 263 



8. Find the minimum value of 



Result, x = , y = , 2=1. 



9. Find the minimum value of x* + y* -f z*, where xyz = c\ 



10. If x, y, z are the angles of a triangle, find the values of 



x, y, z which make sin"*ic sin"y sin p 2; a maximum. 



11. Find the maximum or minimum value of x p y g z r sub- 



ject to the condition Ix + my + nz == a. Hence find the 

 parallelepiped of maximum volume which has for its 

 three edges the axes of x, y, z, and has the intersection 

 of its opposite edges in a given plane. 



12. If a# 2 + bxy + cy^f, and r 2 = # 2 + y, shew that the 



maximum and minimum values of r 2 are given by the 

 equation 



(J 2 - 4ac) / + 4/(a + c) r 2 - 4/ 2 = 0. 



13. Find the maximum value of 



r) 7- 



Result. v = -^? 



, 1 



=v 



14. A given volume V of metal is to be formed into a 

 rectangular vessel ; the sides of the vessel are to be 

 of a given thickness a, and there is to be no lid. De- 

 termine the shape of the vessel so that it may have a 

 maximum capacity. 



Result. If x, y, and z, are the external length, breadth, 

 and depth ; 



/7_ a *\J a? 



