$\"\"$

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The constraints are

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$\"\"$

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The objective function is $\"\"$ .

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Graph :

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Graph the inequalities and shade the required region.

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$\"\"$

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Note : The shaded region is the set of solution points for the objective function.

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Observe the graph,

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Tabulate the solutions of each of two system of inequalities and obtain the intersection points.

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System of boundary equations

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Solution (vertex points)

\"\"

\"\"

\"\"

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$\"\"$

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Find the value of objective function at the solution points.

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At point $\"\"$ , $\"\"$ .

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At point $\"\"$ , $\"\"$ .

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At point $\"\"$ , $\"\"$ .

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The object function have many solutions for different values of $\"\"$ .

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Let $\"\"$ .

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Substituite $\"\"$ in maximum and minimum values.

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$\"\"$ .

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$\"\"$ .

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$\"\"$ .

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Observe the values of $\"\"$ ,

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The maximum value of $\"\"$ is $\"\"$ when $\"\"$ and $\"\"$ .

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The minimum value of $\"\"$ is $\"\"$ when $\"\"$ and $\"\"$ .

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$\"\"$

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The maximum at point $\"\"$ is $\"\"$ .

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The minimum at point $\"\"$ is $\"\"$ .