$\"\"$

\

The constraints are

\

$\"\"$

\

The objective function is $\"\"$ .

\

Graph :

\

Graph the inequalities and shade the required region.

\

$\"\"$

\

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

\

Observe the graph,

\

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

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

\

System of boundary equations

\

\

$\"\"$

\

$\"\"$

\

\

$\"\"$

\

$\"\"$

\

\

$\"\"$

\

$\"\"$

\

Solution (vertex points)

\"\"

\"\"

\"\"

\

$\"\"$

\

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

\

The objective function is $\"\"$ .

\

Let $\"\"$ .

\

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

\

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

\

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

\

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

\

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

\

Let $\"\"$ .

\

The objective function is $\"\"$ .

\

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

\

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

\

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

\

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

\

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

\

Let $\"\"$ .

\

The objective function is $\"\"$ .

\

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

\

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

\

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

\

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

\

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

\

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

\

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

\

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

\

$\"\"$

\

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

\

The objective function is $\"\"$ .

\

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

\

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