# Extreme rays of totally monotone function's cone

A smooth function $f \colon \mathbb{R}^{n}_+ \to \mathbb{R}_{+}$ is said to be totally monotone iff $(-1)^{| \alpha|} \frac{ \partial^{| \alpha |} }{\partial^{\alpha}x} f(x) \geq 0$ for any multi-index $\alpha \in \mathbb{Z}^{n}_{+}$. The totally monotone functions form a convex cone $C$. The task is to show that extreme rays of $C$ are given by the exponential functions $Ae^{-px}$, where $p \in \mathbb{R}^n_{+}$. I have to show this using Lagrange's rule. Please, help me with some ideas.

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What do you mean by "Lagrange rule"? –  Davide Giraudo Feb 1 '12 at 9:14
Lagrange multipliers rule –  Nimza Feb 1 '12 at 13:51
Does Bernstein's theorem help? –  draks ... Mar 30 '12 at 19:06