higher derivatives of functions with given critical points and values

Let $f:B^n\to \mathbb{R}$ be a $d+1$ times continuously differentiable function on the unit ball $B^n$, with ${\max }_{z\in B^n}\mathrm{\parallel }f(z)\mathrm{\parallel }=1$. A well-known fact is that if $f$ vanishes on a set $Z\subset B^n$ with a non-empty interior, then for each $k=1,\dots ,d+1$ the norm of the $k$-th derivative $\mathrm{\parallel }{f}^{(k)}\mathrm{\parallel }$ is at least $M=M(n,k)>0$. A natural question to ask is what happens for other sets $Z$?''. This question was partially answered in [16]-[18]. In the present paper we ask for a similar (and closely related) question: what happens with the high-order derivatives of $f$, if its gradient vanishes on a given set $Σ$? And what conclusions for the high-order derivatives of $f$ can be obtained from the analysis of the metric geometry of the critical values set’’ $f(\mathrm{\Sigma })$? In the present paper we provide some initial answers to these questions.