Example 1.  Two students Ann and Carl work x and y hours per week, respectively.  Together they can work at most 40 hours per week.  According to the rules for part timers Ann can work at most 8 hours more that Carl.  But Carl can work at most 6 hours more than Ann.  There is an extra constraint  .  Determine the region    for these constraints.
1 (a).  If Ann and Carl earn $15 and $17 per hour, respectively, then find their maximum combined income per week.

Solution 1 (a).

Enter the linear function and the constraints.

Graph the region    defined by the constraints.

The solution will occur at one of the vertices of the convex polytope. We now solve for these four points.

Graph the level curves of the objective function.

The solution point for the maximum is the furthest point in the region in the direction of the gradient  .
Find the gradient vector .

Aside.  We are done!

The related problem of finding the minimum is solved in a similar fashion.  All we need to do is check out the function values at the vertices.

(c) John H. Mathews 2005