Past exam of the mathematics course of the University of Cambridge / 2014 / ia / Paper 1 / 11E / i

For $x\ne 0$, put $P(t)={\sum }_{k=0}^{n-1}{f}^{(k)}(0)t^k/k!$ and $K=(f(x)-P(x))/x^n$. The auxiliary function $F(t)=f(t)-P(t)-K{t}^n$ satisfiesRolle's theorem first gives a zero of $F^{\prime }$ strictly between zero and $x$. Apply Rolle's theorem to $F^{\prime }$ between that zero and zero, where $F^{\prime }(0)=0$, to obtain a zero of $F^{\prime \prime }$. Continue in this way. The derivatives through order $n-1$ are continuous, since they are differentiable; hence every application is valid even if $f^{(n)}$ is not continuous. After $n$ applications there is $c$ strictly between zero and $x$ with $F^{(n)}(c)=0$.

But $F^{(n)}(c)=f^{(n)}(c)-n!K$, so $K=f^{(n)}(c)/n!$. Writing $c=\theta x$ yields the Taylor formula with Lagrange remainderThe same construction using repeated applications of Rolle's theorem applies when $x<0$, with intervals ordered from $x$ to zero. For $x=0$ the identity is immediate and any $\theta \in (0,1)$ works. This proves the Taylor theorem with Lagrange remainder using only the allowed theorem and elementary differentiation.