80 DIFFERENTIAL AND INTEGRAL CALCULUS. 



f(x) when # = a, are that /' (x) shall change sign from 

 positive to negative as x increases through the value a; 

 and for a minimum that /' (x) shall change sign from 

 negative to positive as x increases through the value a. 

 In general, the simplest method of finding such values 

 is to observe this change of sign directly, but in some 

 cases, and in especial, when f(x) is what is called an 

 implicit function of cc, this change cannot easily be noted, 

 and a diiferent test to be explained afterwards must be 

 applied. For all the functions which we have commonly 

 to deal with, /' (x) can only change sign by passing 

 through the values or co , otherwise it would be a dis- 

 continuous function, changing its value abruptly as x in- 

 creases gradually. As simple examples of the test, consider 

 the functions (1) x* - 3x + 2, (2) (x - a)*. 



(1) f(x] = x 3 - 3x + 2, /' (x) = 3 (x + 1) (x - 1), when x 

 has any value between GO and - 1, x + 1, x 1 are both 

 negative and f'(x] positive, or f(x) increases with x\ but 

 when x passes the value - 1, and before it becomes so 

 great as 1, x+ 1 is positive and # 1 negative; therefore 

 f (x) is negative and f(x) is decreasing. When x has 

 passed the value I,/' (x) is again positive, and f(x) again 

 increases with x. Hence /( 1) or 4 is a maximum and 



/(I) or a minimum value of f(x). 



(2) f(x) = (x-a)*,f ( X ) = }. (*-)-*, when x<a,f (x) 

 is negative ; and when x> a, f'(x) is positive ; hence, /(a?) 

 decreases as x increases from co to a, and then increases, 

 and f(a) or is a minimum value of f(x). 



To illustrate these results geometrically, draw the curves 

 represented by the equations y = x 3 - 3x + 2, y = (x - aft 

 respectively, i. e. to every distance x measured from along 

 the fixed straight line 0#, draw at right angles from its 

 extremity a length y = x 3 - 3x -t- 2 in (1), or to (#-)* in 

 (2). In(l) 0^1 = 1, 05= -1, 05 = 2,56 = 4,50=-!, 

 and the curve is somewhat as in fig. 12, so that y has the 

 maximum value 5Q = 4, and its minimum value at A, 

 although on the branch beyond A there are an infinite number 



