Relationships Which Exist Between Functions Of A Single Variable And Their Derivatives Or Differentials Of Various Orders. Use Of These Differentials In The Study Of Maxima And Minima

Suppose that the function f(x) vanishes for the particular value $$x_0$$ of the variable $$x. \ $$ Consider, moreover, that this same function and its successive derivatives, up to that of order n, are continuous in the vicinity of the particular value in question, and that the continuity remains valid for each of them between the two limits $$x=x_0, x=x_0+h. \ $$ Equation ( 6 ) of the seventh lecture will become $$\begin{aligned} f(x_0+h)=f(x_0)+hf^{\prime }(x_0+\theta h)=hf^{\prime }(x_0+\theta h), \end{aligned}$$ .