3 (30 points) John has $2,000 to invest in three funds A, B and C. Fund A offers a return of 1% and has a low risk. Fund B offers a return of 4% and has a medium risk. Fund C offers a return of 5% but has a high risk. To be on the safe side, John invests no more than $500 in C and at least twice as much as in A than in B. Assuming that the rates hold till the end of the year, what amounts should he invest in each fund in order to maximize the year end return? Let x be the amount invested in A, y the amount invested in B, and z the amount invested in C. This problem is formulated as follows: Maximize 0.01x + 0.04y+ 0.05z Subject to x+ 3+ z=2000 エー 2y 0 << 500 x, y, 0 Substituting z=2000-(x+y) into the linear program above, we reformulate it to be a bivariate model with x and y. Maximize Subject to -0.04x- 0.01y+100 x+ y≤2000 x + y≥ 1500 2y> 0 x, y 0 a: Draw the feasible region of the bivarate linear program. b: Find an optimal solution of the bivarate linear program, and calulate the optimal year end return. c: Write the dual program of the bivarate linear program. d: Compute an optimal dual solution using the complementary slackness prop- erty. e: Complete the following three sentences using the results obtained in Parts a-d: If John decides to invest no more than $400 in Fund C, it is estimated that the optimal year end return that you caculate in Part B (increases/decreases) by dollars at (least most). If Fund A now offers a return of 2%, it is estimated the optimal year end return (increases/decreases) by dollars at (least most). If Fund C now offers a return of 4.5%, it is estimated the optimal year end return (increases/decreases) by dollars at (least most).