Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 1 / 11E (Analysis)

The mean value theorem says that for continuous $f$ on $[a,b]$, differentiable inside, $f(b)-f(a)=f^{\prime }(c)(b-a)$ for some $c$. Applied to $\log$ on $[b,a]$,which gives $(a-b)/a<\log (a/b)<(a-b)/b$.

The ordinary mean value theorem gives ${\Delta }_{h}f(a)=h{f}^{\prime }(b_1)$ for some $b_1\in (a,a+h)$. More generally, applying it to the difference of two translated functions shows inductively thatfor some $b_k\in (a,a+kh)$. Taking $k=n$ proves the result.

No. The Darboux theorem says that every derivative has the intermediate-value property. Choose $ϵ<1/3$. In a deleted neighborhood of $a$, the assumed limit puts $\varphi (x)$ within $ϵ$ of $\varphi (a)+1$. Darboux's theorem applied between $a$ and any such $x$ would require values near $\varphi (a)+1/2$, which that neighborhood excludes. This contradiction shows that $\varphi$ cannot be a derivative.