Answer
.The values of the objective function at points \[\left( 2,6 \right),\left( 6,3 \right),\left( 8,0 \right),\text{ and}\left( 2,0 \right)\] are \[18,24,24,\text{ and 6}\text{.}\]The maximum value of the objective function is 24, and a minimum value of the objective function is 6.
Work Step by Step
From the given graph below, the coordinate of all the corner point are as follow:
\[\left( 2,6 \right),\left( 6,3 \right),\left( 8,0 \right),\text{ and}\left( 2,0 \right).\]
The value of the objective function at a point\[\left( 2,6 \right)\]is:
\[\begin{align}
& z=3\times 2+2\times 6 \\
& =6+12 \\
& =18
\end{align}\]
The value of the objective function at a point\[\left( 6,3 \right)\]is:
\[\begin{align}
& z=3\times 6+2\times 3 \\
& =18+6 \\
& =24
\end{align}\]
The value of the objective function at a point\[\left( 8,0 \right)\]is:
\[\begin{align}
& z=3\times 8+2\times 0 \\
& =24+0 \\
& =24
\end{align}\]
The value of the objective function at a point\[\left( 2,0 \right)\]is:
\[\begin{align}
& z=3\times 2+2\times 0 \\
& =6
\end{align}\]
Therefore,the values of objective function at points \[\left( 2,6 \right),\left( 6,3 \right),\left( 8,0 \right),\text{ and}\left( 2,0 \right)\] are \[18,24,24,\text{ and 6}\text{.}\]The maximum value of the objective function is 24, and a minimum value of the objective function is 6.
