TAYLOR'S THEOREM. 71 ' 



all positive values of x, it follows by the present Article 

 that y is always increasing so long as x is positive ; but 

 y = when x = ; therefore y is positive for all positive values 

 of x. 



Similarly we can shew that x log (1 + x) is positive for 

 all positive values of x. 



90. A function of a variable is said to be continuous be- 

 tween certain values of the variable when it fulfils the follow- 

 ing conditions : the function must have a single finite value 

 for every value of the variable, and the function must change 

 gradually as the variable passes from one value to the other, 

 so that corresponding to an indefinitely small change in the 

 variable there must be an indefinitely small change in the 

 function. 



91. Suppose < (x} a function which vanishes when x = a 

 and when x = b, and is continuous between those values. 

 Suppose also that <j> (x} is continuous between those values. 

 Then <' (x) will vanish for some value of x between a and b. 



For (f> (x} cannot be always positive between those values, 

 for then < (x} would be constantly increasing as the variable 

 increased from the lower value to the higher (Art. 89), which 

 is inconsistent with the supposition that <f> (x} vanishes at the 

 two specified values. Similarly <f>' (x) cannot be always nega- 

 tive. Hence <' (x} must change from positive to negative or 

 from negative to positive between the assigned values ; and 

 since it is continuous it cannot become infinite and must 

 therefore pass through the value zero. 



If a denote some constant quantity, such expressions as 

 f (a),/" (a), ... may occur in our investigations, the meaning 

 to be attached to them being that f(x) is to be differentiated 

 once, twice, . . . with respect to x, and in the result x changed 

 into a. 



We can now demonstrate Taylor's Theorem. The proof 

 which we give in the next Article is due to Mr Homersham 

 Cox; it was published by him in the 6th volume of the 

 Cambridge and Dublin Mathematical Journal, and subse- 

 quently in his Manual of the Differential Calculus. 



