$(4)$ Financial Application $$ financial advisor at Diehl investments identifies two companies that are likely candidates for a takeover in the near future. Eastern Cable, is a leading manufacturer of flexible cable systems used in the construction industry and ComSwitch is a new firm specializing in Digital switching systems. Eastern Cable is currently trading for $$40$ a share and ComSwitch is currently trading at $$25$ a share.

If the takeovers occur, the financial advisor estimates the price of Eastern will go to $$55$ a share and ComSwitch to $$43$ a share. At this point in time, the advisor has identified ComSwitch as the higher risk alternative.

Assume that a client shows a willingness to invest a maximum of $$50000$ in the two companies. The client wants to invest at least $$15000$ in Eastern and at least $ $10000$ in ComSwitch. Because of the higher risk associated with ComSwitch the financial advisor has recommended that at most $$25000$ be invested with ComSwitch.

$($a$)$ Formulate a linear programming model that can be used to determine the number of shares of Eastern and the number of shares of ComSwitch that will meet the investment constraints and maximize the total return of the investment.

$($b$)$ Graph the feasible region and determine the coordinates of each extreme point.

$($c$)$ Are there any alternative optimal points? If yes, what is the corresponding return of the investmont?