proof of calculus theorem used in the Lagrange method

Let $f(\mathbf{x})$ and $g_{i}(\mathbf{x}),i=0,{\ldots},m$ be differentiable scalar functions; $\mathbf{x}\in R^{n}$ .

We will find local extremes of the function $f(\mathbf{x})$ where $\nabla f=0$ . This can be proved by contradiction:

$\nabla f\neq 0$

$\Leftrightarrow\exists\epsilon_{0}>0,\forall\epsilon;0<\epsilon<\epsilon_{0}:f% (\mathbf{x}-\epsilon\nabla f)<f(\mathbf{x})<f(\mathbf{x+\epsilon\nabla f})$

but then $f(\mathbf{x})$ is not a local extreme.

Now we put up some conditions, such that we should find the $\mathbf{x}\in S\subset R^{n}$ that gives a local extreme of $f$ . Let $S=\bigcap_{i=1}^{m}S_{i}$ , and let $S_{i}$ be defined so that $g_{i}(\mathbf{x})=0\forall\mathbf{x}\in S_{i}$ .

Any vector $\mathbf{x}\in R^{n}$ can have one component perpendicular to the subset $S_{i}$ (for visualization, think $n=3$ and let $S_{i}$ be a flat surface). $\nabla g_{i}$ will be perpendicular to $S_{i}$ , because:

$\exists\epsilon_{0}>0,\forall\epsilon;0<\epsilon<\epsilon_{0}:g_{i}(\mathbf{x}% -\epsilon\nabla g_{i})<g_{i}(\mathbf{x})<g_{i}(\mathbf{x}+\epsilon\nabla g_{i})$

But $g_{i}(\mathbf{x})=0$ , so any vector $\mathbf{x}+\epsilon\nabla g_{i}$ must be outside $S_{i}$ , and also outside $S$ . (todo: I have proved that there might exist a component perpendicular to each subset $S_{i}$ , but not that there exists only one; this should be done)

By the argument above, $\nabla f$ must be zero - but now we can ignore all components of $\nabla f$ perpendicular to $S$ . (todo: this should be expressed more formally and proved)

So we will have a local extreme within $S_{i}$ if there exists a $\lambda_{i}$ such that

$\nabla f=\lambda_{i}\nabla g_{i}$

We will have local extreme(s) within $S$ where there exists a set $\lambda_{i},i=1,{\ldots},m$ such that

$\nabla f=\sum\lambda_{i}\nabla g_{i}$

Title	proof of calculus theorem used in the Lagrange method
Canonical name	ProofOfCalculusTheoremUsedInTheLagrangeMethod
Date of creation	2013-03-22 13:29:51
Last modified on	2013-03-22 13:29:51
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	6
Author	mathcam (2727)
Entry type	Proof
Classification	msc 15A18
Classification	msc 15A42