24G EXAMPLES OF MAXIMA AND MINIMA. 



17. Shew that -7 5 ^ is a maximum when x = - , 



a 

 c 



y -' 



9 a 



18. Shew that a:e irhtsiny has neither a maximum nor a mini- 



mum. 



19. Find the minimum value of x + y + z, subject to the 



condition 



?+M-i. 



x y z 



Result. When -^- = -^ L = -^- = *Ja + Jb + Jc. 

 V \o v c 



20. Find the minimum value of afy q z r subject to the same 



condition as in the preceding Example. 



Result. When -^ = ^= = p + q + r . 

 a b c 



21. Having given the three sides of a triangle, find a point 



within it, such that, if perpendiculars be drawn from 

 it to the sides, their continued product shall be a 

 maximum. Shew that straight lines joining this point 

 with the corners of the given triangle will divide it 

 into three equal triangles. 



22. Find the maximum value of scyz subject to the con- 



dition 



j, ,, abc 

 Result. . 



23. Determine a point within a triangle, such that the sum 

 of the squares on the distances from the three sides is 

 a minimum. 



Result. If p, q, r, be the perpendiculars on the sides 

 a, b, c, respectively, then 



p _q _r _ 2 area of triangle 

 a~b~c ~ a 2 + b' + c* * 



