# 04-Some-proves-and-Applications

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from scipy.optimize import minimize

### Question 1

Consider the two problems and Show that if and only if

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=