( 193 ) 



CHAPTER XIII. 



MAXIMA AND MINIMA OF FUNCTIONS OF ONE VARIABLE. 



209. SUPPOSE < (x) to denote a certain function of x, 

 and that while the variable x changes gradually from one 

 definite value to another, <j> (x) changes in such a manner 

 that it is sometimes increasing and sometimes decreasing. 

 There must then be certain values of x, for which $ (x) begins 

 to decrease, having previously been increasing, or begins to 

 increase, having previously been decreasing. In the former 

 case, < (x) has a greater value for the particular value of x 

 than it has for adjacent values of x, and is said to have 

 a maximum value. In the latter case, < (x) has a less value 

 for the particular value of x than it has for adjacent values 

 of x, and is said to have a minimum value. Hence, these 

 terms maximum and minimum are not used to denote the 

 arithmetically greatest and least values which a function can 

 assume; for it appears from the above explanation that a 

 function may have several maxima and minima values, and 

 that some particular minimum may be greater than some 

 particular maximum. 



210. DEFINITION. If as x increases or decreases from 

 the value a through a finite interval, however small, $ (x} 

 is always less than (f> (a), then < (a) is called a maximum 

 value of < (x) ; if </> (x) is always greater than $ (a), then 

 < (a) is called a minimum value of < (#). 



211. Rule for discovering maxima and minima values. 



Let $ (x) denote any function of x. By Art. 92, we 

 have 



< (x + h) = (j> (x) + hf (x) + ^ <f>" (x + Bh). 



ti 



T. D. C. O 



