# proof of calculus theorem used in the Lagrange method

Let $f(\mathbf{x})$ and ${g}_{i}(\mathbf{x}),i=0,\mathrm{\dots},m$
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^{}:

$$\nabla f\ne 0$$ |

$$ |

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:

$$ |

But ${g}_{i}(\mathbf{x})=0$, so any vector $\mathbf{x}+\u03f5\nabla {g}_{i}$ 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 ${\lambda}_{i},i=1,\mathrm{\dots},m$ 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 |