SPECIAL APPLICATIONS OF ANALYSIS 



153 



Minimum, 



a + 



a- 2V P 



7.203 If for x = a, f'(a) = o and }"(a) = o, in order to determine whether 

 y = f(a) is a maximum or minimum it is necessary to form the higher differential 

 coefficients, until one of even order is found which does not vanish for x = a. 

 y = f(a) is a maximum or minimum according as the first of the differential 

 coefficients, f"(a), / iv (a), /^(a), . . . . . of even order which does not vanish 

 is negative or positive. 



7.210 Functions of Two Variables. F(x, y) is a maximum or minimum for the 

 pair of values of x and y that satisfy the equations, 



dF 



dF 



and for which 



dx 0> dy 



(dxdy) dx 2 



-^- n <o. 



dxdy/ dx* 



are negative for this pair of values of x and y, F(x, y) is 

 a maximum. If they are both positive F(x, y) is a minimum. 



If both ^ and 5 



7.220 Functions of n Variables. For the maximum or minimum of a function 

 of n variables, F(XI, x z , ), it is necessary that the first derivatives, 



1 F 



[' dxj 



\ Tf 



. all vanish; and that the lowest order of the higher deriv- 



OX n 



atives which do not all vanish be an even number. If this number be 2 the 

 necessary condition for a minimum is that all of the determinants, 



D k 



where 



fllflZ /I* 



fzi /22 /2fc 



/a/' 



A-2 



.n, 



