Let $f$ be a smooth function on an open interval $I$. Let $x_{0}\in I$ and suppose that
1) $f(x_{0})=0$
2) $|f^{(i)}(x_{0})|\sim N$ for some large $N$ for all $i\geq 1$ ( meaning that all derivatives of $f$ are of size about $N$ at $x_{0}$; more precisely, there exist two constants $c_{1},c_{2}>0$ such that $$c_{1}N\leq |f^{(i)}(x_{0})|\leq c_{2}N $$ for all $i\geq 1$.
Question:
How long can any of these derivatives stay of size $N$ ?
I mean what is the largest $\delta$ such that, say, $|f^{\prime}(x)|\sim N$ on $]x_{0}-\delta,x_{0}+\delta[$ ?
For simplicity, assume $|f^{(i)}(x_{0})|=1$ for all $i\geq 1$. Find the largest $\delta$ such that $|f^{\prime}(x)|\geq \frac{1}{2}$ on $]x_{0}-\delta,x_{0}+\delta[$ ?
Let $\phi$ be a non negative $C^\infty$ function with compact support on $[-1,1]$ and such that $\phi(x)=1$ if $|x|\le1/2$. Given $\epsilon>0$ let $$ f_\epsilon(x)=(e^x-1)\phi(x/\epsilon). $$ Then $f_\epsilon(x)=e^x-1$ if $|x|\le\epsilon/2$, $f_\epsilon(0)=0$ and $f^{(i)}(0)=1$ for all $i\ge1$. But $f$ and all its derivatives are equal to $0$ when $x=\pm\epsilon$. Since $\epsilon$ is arbitrary, we see that there is no $\delta>0$ depending only of the values of the derivatives of $f$ at $x=0$ such that $|f'(x)|\ge1/2$ on $(-\delta,\delta)$.