82 DIFFERENTIAL AND INTEGRAL CALCULUS. 



. even integer . f 



, ^.e. JJ-^-T j that factor itself can never 

 I ' odd inteer ' 



n , .. JJ 



2q + I ' odd integer 



change sign, and, therefore, f (x) will not change sign 

 as x passes through that particular value, or there will be 

 neither maximum nor minimum corresponding to that 

 factor. Such values being struck out of the list, arrange 

 the remainder in order of increasing magnitude a, 5, c... ; 

 observe the sign of /' (x) when x<a, and, therefore, x a 

 negative. If this sign be positive,/' (x} will change from 

 positive to negative as x increases through a, and /(a) will 

 be a maximum, f(b) a minimum, f(c) a maximum, and 

 so on, until all the reserved factors have been taken 

 account of. Thus, suppose 



f , M 

 J ( x ) 



(a: + 4) 



so that / (x) vanishes when x = - 3, - 2, - 1, 0, 1, 2, 3, 

 and is GO when x = 4. Here the index of the factor 

 x 4 3 is 4, of x + 1 is f , and of x 2 is 2, and none of 

 these factors can change sign. The remaining critical 

 values of x are - 4, 2, 0, 1, 3, and when x is between 

 - co and 4,/' (x) has 5 negative factors and is therefore 

 negative. Hence # = -4 gives f(x) a minimum, x = 2 

 gives f(x) a maximum, /(O) is a minimum, /(I) a maxi- 

 mum, and/ (3) a minimum. 



The above method gives the most satisfactory general 

 rule for determining and distinguishing maximum -and 

 minimum values. In a very large proportion of geome- 

 trical applications, the nature of the question tells us at 

 once whether the result is a maximum or a minimum, 

 especially when there is only one solution of either sort. 

 For instance, if PQ is a straight line drawn through a 

 given point A and terminated by two given straight lines 

 Ox) Oy, and the maximum or minimum length of PQ 

 is required, it is manifest that when the straight line is 

 drawn parallel to either Ox or Oy the length of the line 



