I had this question. Let's say that there is a function and it's $n$ times continuously differentiable. Take a random $a$ from $\Bbb R$ and $\delta$>0. Prove that there are polynomials $p$ and $q$ with degree lower than or equal to $n$ and a $m$ from $\Bbb R$ so that: $p(x)\le f(x)\le q(x)$ and $q(x)-p(x)\le m(|x-a|)^{n}$ for $x$ in $[a,a+\delta[$.
I thought maybe I can use Taylor. Is there someone who can help me with this problem?
Fix $a$ and $\delta>0$, and define \begin{align} P(x)=\sum_{k=0}^{n-1}\frac{f^{(k)}(a)}{k!}(x-a)^k\,. \end{align} By Taylor's theorem, for each $x\in[a,a+\delta]$, there exists $\xi_x\in[a,x]$ such that \begin{align} f(x)=P(x)+\frac{f^{(n)}(\xi_x)}{n!}(x-a)^n\,. \end{align} Let $A$ and $B$ be the minimum and maximum of $f^{(n)}$ on $[a,a+\delta]$ respectively. Put \begin{align} p(x)&=P(x)+A(x-a)^n/n!\,, \\ q(x)&=P(x)+B(x-a)^n/n!\,. \end{align} Then $p$ and $q$ have the desired properties.