DIFFERENTIAL AND INTEGRAL CALCULUS. 83 



is infinite, hence there must be a minimum length in some 

 intermediate position. 



Another criterion for distinguishing maximum and mini- 

 mum values is found as follows : f(x) being a function of 

 the independent variable a?, f(a) is said to be a maximum 

 value of /(a?), if it is greater than all adjacent values of 

 /(a?); i.e. if we put a + h or a h for a, then the results 

 are both less than /"(a) when h is taken sufficiently small. 



Hence, for a maximum 



/(a + h) -f(a), and /( - h} -/(), 



are both negative ; and similarly for a minimum are both 

 positive. But if f(x) and its differential coefficients be 

 finite for values of x between a and a + Ti, 



f(a + h) =/() + hf (a) + /" (a + Oh] ; 



or f(a + A) _ /(a) = A/ (a) + /' ( + ^) ; 



/( - A) -/(a) = - A/ (a) + ~ /" (a - <?A). 



Now, if/' (a) be finite, since we can by diminishing h make 

 the second term of each of these numerically less than the 

 first, these two expressions must have opposite signs when 

 h is taken sufficiently small. Hence, there can be neither 

 maximum nor minimum unless /' (a) = 0. If this be so, 

 f(a + h)f(a)i and f(a- h) -/() will, when h is suffi- 

 ciently small, have the same sign as /" (a). But for a 

 maximum both must be negative ; therefore for a maximum 

 value /(), we must have /' (a) = 0, and /" (a) negative. 

 So, for a minimum, /' (a) = 0, /" (a) = a positive quantity. 

 If/" (a) = 0, and in general if all the differential coefficients 

 up to the (n- l) th vanish, while/" (a) is finite, we have 



