196 MAXIMA AND MINIMA VALUES 



<j>(x) changes from positive to negative if < n (a) be negative, 

 that is, if $() be a maximum ; and <' (x) changes from nega- 

 tive to positive if <f> n (a) be positive, that is, if </> (a) be a 

 minimum. This suggests another form for the definition 

 of maxima and minima values and for the investigation 

 of the conditions of their existence which we give in the 

 next Article. 



214. DEFINITION. If as x varies through any finite in- 

 terval, however small, < (x) increase until x = a and then 

 decrease, <j> (a) is called a maximum value of <f> (x) ; if <f> (x) 

 decrease until x = a and then increase, <f> (a) is called a mini- 

 mum value. 



By Art. 89, if the differential coefficient of a function 

 be positive that function increases with the variable, and if 

 the differential coefficient be negative the function decreases 

 as the variable increases. Hence, as x increases <f>'(x) must 

 change from positive to negative when x = a, if tf> (a) be a 

 maximum, and from negative to positive if <f> (a) be a minimum. 

 But a function can only change its sign by passing through zero 

 or infinity. Hence, we must find the values of x that make 



=0, 



or <f> (a?) = oo ; 



and if as a; passes through any one of these values <' (x) 

 changes its sign, we have for that value of # a maximum 

 or minimum value of < (x), according as, when x increases, the 

 change is from positive to negative or from negative to positive. 



Example (1). Suppose (f> (x} =x 3 9ce s + 24# - 7, 

 then $ (x) = 3 (cc 2 - 6x + 8), 



<f>" (a?) = 6 (a; -3). 



If we put $' (#) = 0, we obtain x = 2, or x = 4 ; 

 when x = 2, <" (x) is negative, 



when x = 4, <" (x) is positive. 



Therefore when x = 2, < (x) has a maximum value, and 



when x = 4, < (x) has a minimum value. 

 Example (2). Let $ {x} = e x + e~* + 2 cos x ; 

 therefore <J>' (x} =e x -e~ x -2 sin x, 



-2 cos x, 



