OF A FUNCTION OF ONE VARIABLE. 197 



<f>'"(x)=e x -e~* + 2sino?, 



I? a = 0, -WQ have 0'(a;)=0, $"(x} = Q, </>"'(a;)=0, and 

 <>"" (x) = 4. Hence, < (x) is a minimum when x = 0. 



It may be easily shewn that x = is the only value of x 

 for which <j> (#) vanishes ; for 



o oin <r 9 J -y '- JL 



i. bin *c ^ -> cc -j- ... 



I U [A 



f 9 



therefore <' (a;) = 



All the terms in <' (a;) being of the same sign, </>' (x) can never 

 vanish except when x = 0. 



Example (3). Suppose -7- = a; (x I) 2 (a? 3) 3 , for what 



values of x will w be a maximum or minimum? In this 

 Example the method of Art. 214 is preferable. When x is 



negative -=- is positive ; when x is positive and less than 



unity, -r- is negative. Hence j- changes from positive to 

 ctx ctx 



negative as x passes through the value 0, and x = makes u 



CLtL 



a maximum. When x = 1, -7- vanishes ; it does not how- 



dx 



ever change its sign, but continues negative until x = 3, and 

 after that it is positive. Hence, when x = 1, u has neither a 

 maximum nor minimum value, but has a minimum value 

 when x = 3. 



Suppose that in the Example last given we merely wish 

 to ascertain if x = gives a maximum or minimum value to u, 

 and that we are required to proceed according to the method 

 of Art. 211 : we have 



