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.
Extreme rays of totally monotone function's cone
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optimization
convex-analysis
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0Does [Bernstein's theorem](http://en.wikipedia.org/wiki/Bernstein%27s_theorem_on_monotone_functions) help? – 2012-03-30