# 04-Some-proves-and-Applications

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### Question 1

Consider the two problems \begin{align} \text{Maximize } f(x_1,\dots,x_n) \text{ subject to } (x_1,\dots,x_n)\in \Omega\\ \end{align} and \begin{align} \text{Minimize } -f(x_1,\dots,x_n) \text{ subject to } (x_1,\dots,x_n)\in \Omega \end{align} Show that \begin{align} f(x_1^*,\dots,x_n^*)=max\{f(x_1,\dots,x_n):(x_1,\dots,x_n) \in \Omega \}\\ \end{align} if and only if \begin{align} -f(x_1^*,\dots,x_n^*)=min\{-f(x_1,\dots,x_n):(x_1,\dots,x_n) \in \Omega \}\\ \end{align}

With out loss of generality we assume these problems have unique solutions.

Assuming:

We have that:

which is:

The reverse also holds:

Assuming:

We have that:

which is:

### Question 2

An import car company has warehouses in cities $A$,$B$ and $C$ and supplies four different dealers $D_1$,$D_2$,$D_3$ and $D_4$. The cost in dollars of transporting a car from a given warehouse to a given dealer is found in the following table. Currently, $D_1$ needs $100$, $D_2$ needs $50$, $D_3$ needs $65$, and $D_4$ needs $75$ cars. How should the cars be shipped in order to minimize the shipping costs?

C=$% $

X=$% $

L=$% $

P=$% $

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