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\textbf{Applications of Partial Differentiation}

\textit{\textbf{Extremes within restricted domains}}
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\textbf{Question}

$F(x,y,z)=2x+3y+4z$

Minimize $F$, subject to
\begin{eqnarray*}
x & \ge & 0\\
y & \ge & 0\\
z & \ge & 0\\
x + y & \ge & 2\\
y + z & \ge & 2\\
x + z & \ge & 2
\end{eqnarray*}


\textbf{Answer}

The constraint region has vertices $(1,1,1)$, $(2,2,0)$, $(2,0,2)$ and
$(0,2,2)$.

This gives
$$F(1,1,1)=9, \ \ \ F(2,2,0)=10,\ \ \ F(2,0,2)=12, \ \ \ F(0,2,2)=14$$
And so the minimum value of $F$ under the given constraints is $F=9$.

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