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\textbf{Applications of Partial Differentiation}

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

$Q(x,y)=2x+3y$

Maximize $Q$, subject to
\begin{eqnarray*}
x & \ge & 0\\
y & \ge & 0\\
y & \le & 5\\
x+2y & \le & 12\\
4x + y & \le & 12
\end{eqnarray*}


\textbf{Answer}

Maximize $Q(x,y)=2x+3y$, in the region of constraint pictured below.

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Notice that if a point satisfies $y \le 5$ and $4x+y\le12$ then it
also satisfies $x+2y \le 12$. 

$y=5$ and $4x+y=12$ meet at $\displaystyle \left ( \frac{7}{4},
5\right)$, and so the maximum value of $Q(x,y)$ under the constraints
is
$$Q \left ( \frac{7}{4} , 5 \right ) = \frac{7}{2} + 15 = \frac{37}{2}.$$

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