$1.$ A mutual fund manager is trying to decide on how to divide up $$12$ million between domestic

and foreign stocks. Domestic stocks have been returning $11\%$ per year and foreign $17\%.$

The fund manager wants to maximize the annual return from the investments. However the

manager also wants to be cautious. No more than $$10$ million of the fund should go into

domestic stocks and no more than $$7$ million into foreign. Also at least half as much should

be invested in foreign as domestic and at least half as much in domestic as foreign to maintain

some balance.

$($a$)$ Formulate a linear program to solve the problem.

$($b$)$ Draw the feasible region of the linear program and identify all the corner points of the

feasible region.

$($c$)$ Solve the linear program graphically.