198 



MAXIMA AND MINIMA VALUES 



d*u 

 when x = the first term in -7-5 is negative, and the other two 



CLdC 



terms vanish since they both have x as a factor. Hence we 

 need not have expressed them, but might have put 



d*u 



-j-j = (a; I) 2 (a; 3) 3 + terms vanishing when x = 0. 



dJo 



This remark should be carefully noticed, because in Exam- 

 ples like the above we are saved the trouble of writing down 

 superfluous terms. 



Example (4). The following Example will introduce the 

 reader to considerations by which the process for finding 

 maxima and minima values may sometimes be abbreviated. 



Through a given point P a 

 straight line is drawn, meeting 

 the axes Ox and Oy at A and B 

 respectively : find the least length 

 this straight line can have. 



Let OM=a,MP=b, PA = 0. ~ 



Then 



PA = ~ 



cos$' 



7 



Put u = -. H -| a , and we have to find the least value of v. 

 sin o cos o 



b cos 9 a sin 



Now 



du 

 dd 



cos 2 6 ' 



therefore -^ vanishes only when tan 6 = A / - . 

 dd V a 



From the figure it appears that by making either as 

 small as we please, or as nearly equal to a right angle as 



