$\"\"$

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The objective function is $\"\"$ .

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The constraints are

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$\"\"$

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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 $\"\"$ such that objective function has maximum value at $\"\"$ by trail and error method.

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Let $\"\"$ .

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The objective function is $\"\"$ .

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Substitute $\"\"$ and find maximum and minimum values.

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

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

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

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Observe that the objective function is neither a maximum nor a minimum value, hence $\"\"$ .

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Let $\"\"$ .

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The objective function is $\"\"$ .

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Substitute $\"\"$ and find maximum and minimum values.

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

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

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

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Observe that the objective function has maximum value from $\"\"$ to $\"\"$ , hence $\"\"$ .

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Let $\"\"$ .

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The objective function is $\"\"$ .

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Substitute $\"\"$ and find maximum and minimum values.

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

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

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

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Observe that the objective function has maximum value at $\"\"$ , hence $\"\"$ .

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Therefore, the value of the $\"\"$ is $\"\"$ .

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The objective function is $\"\"$ .

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The maximum value of $\"\"$ is $\"\"$ units at $\"\"$ .

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The minimum value of $\"\"$ is $\"\"$ units at $\"\"$ .

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$\"\"$

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The value of the $\"\"$ is $\"\"$ .

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The objective function is $\"\"$ .

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The maximum value of $\"\"$ is $\"\"$ units at $\"\"$ .

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The minimum value of $\"\"$ is $\"\"$ units at $\"\"$ .