OF A FUNCTION OF ONE VARIABLE. 199 



we please, the straight line AB may be made as great as we 



7T 



please. Also, as 6 varies from to - , there must be some 



2 



value of 9 which gives to the straight line AB the least length 

 it can have, and this least length of AB will satisfy the defi- 



nition of a minimum length. And as -^ for a value of be- 



du 



tween and -- can never change its sign except when 



m 



3 /b 

 tan = ./ - , this must be the value of 6 that gives the least 



length we are seeking. 



This value of gives for the least length the value 



In this Example it is easy to see from the value of -^, 



that it does change sign from negative to positive when 

 increases and passes through the value assigned ; but in more 

 complicated questions it is often advisable to shew in the 

 manner above exemplified, that a maximum or minimum 

 must necessarily exist, and then we are saved the trouble of 

 examining if the differential coefficient of the function changes 

 sign when it vanishes. 



215. If u be a function of x we have shewn that -7- = 



ax 



is the equation from which we are to find values of x which 

 make u a maximum or a minimum. If then between two 



assigned values of x there exists no value which makes ~ 



ctoo 



vanish, we conclude that there is no maximum or minimum 

 value of u between those assigned values of a;; so that u 

 either continually increases or continually decreases as x 

 changes from the less to the greater of the assigned values. 

 This principle has already been noticed in Art. 89, but its 

 importance and its natural connexion with the subject of the 

 present Chapter lead us to draw attention to it again. 



For example, suppose 



u 2x tan -1 a; log {x 



