( 22G ) 



CHAPTER XV. 



MAXIMA AND MINIMA VALUES OF A FUNCTION OF TWO 

 INDEPENDENT VARIABLES. 



227. DEFINITION. A function < (x, y) of two indepen- 

 dent variables is said to have a maximum value when 

 <f> (x + h, y + k) is less than </> (x, y] for all values of h and k. 

 positive or negative, comprised between zero and certain 

 tinite limits however small. The function is said to have a 

 minimum value when <J> (x + h, y + k) is greater than $ (x, y) 

 for all such values of h and k. 



228. To investigate the conditions that a function of two 

 independent variables may have a maximum or minimum 

 value. 



Let u = <j> (x, y), 



v = <f>(x + 6!i,y + 6k] ; 

 then, by Art. 226, 



du , du 



n 2 



where -^nr i* T~+ 



~ 



nr T~ j~r ^r* - 



2_ ( dx~ dx dy dy ) 



Now, if h -J- + k -j- be not zero, by taking h and k suf- 



ficiently small, we can always make R less than h -7- -f k -^- , 

 and hence the sign of <f> (x + h, y + k) <f> (x, y) will depend on 

 that of h -T- + k -j- , and will therefore change by changing 



that of h and k ; it is impossible then that < (x, y} can have 

 a maximum or minimum value unless 



** + *-* 



ax ay 



