Let $p\in \mathbb{P}_2$ such that $p(i)=\ln (i)$, $i=2,3,4$. I want to show that for the error $\epsilon(x)=f(x)-p(x)$, $x\in [2,4]$, it holds that $$-\frac{1}{64}\leq \epsilon (3,5)\leq -\frac{1}{512}$$ Could you give me a hint how we could find these inequalities?
Do we have to use the error of the Taylor polynomial?
Consider $$R(t)=ϵ(x)(t-2)(t-3)(t-4)-ϵ(t)(x-2)(x-3)(x-4).$$ Then $R$ has $4$ roots $2,3,4,x$, by Rolle's theorem $R'$ has $3$ roots, $R''$ has $2$ roots, and $R'''$ has one root $\xi$. Thus $$ 0=R'''(ξ)=6ϵ(x)-ϵ'''(\xi)(x-2)(x-3)(x-4) $$ and as $p$ is quadratic, $ϵ'''(\xi)=f'''(\xi)$ so that $$ ϵ(x)=\frac{f'''(\xi)}6(x-2)(x-3)(x-4) $$ and inserting $$ ϵ(3.5)=-\frac{f'''(\xi)}{16} $$ for some $\xi\in(2,4)$. Now insert upper and lower bounds for the third derivative.