Jeff is willing to invest $5000 in buying shares and bonds of a company to gain maximum returns. From his past experience, he estimates the relationship between returns and investments made in this company to be:

R = –2S2 – 9B2 – 4SB + 20S + 30B.

where, R = total returns in thousands of dollars S = thousands of dollars spent on Shares B = thousands of dollars spent on Bond Jeff would like to develop a strategy that will lead to maximum return subject to the restriction provided on amount available for investment.

a. What is the value of return if $3,000 is invested in shares and $2,000 is invested bonds of the company? b. Formulate an optimization problem that can be solved to maximize the returns subject to investing no more than $5,000 on both share and bonds. c. Determine the optimal amount to invest in shares and bonds of the company. How much return will Jeff gain? Round all your answers to two decimal places.

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With $3000 being invested in shares and $2000 being invested in bonds, we can simply substitute these values into the returns function (remembering that the variables are defined as thousands of dollars).
?
R = – 2S2 – 9B2 – 4SB + 20S + 30B
?
= – 2(32) – 9(22) – 4(3)(2) + 20(3) +(30(2)
?
= 42
Return of $42,000 will be realized with this allocation of the investment.
?
 b. and c.
We simply add an investment constraint and non-negativity constraint to the return function that is to be maximized.
?
Max – 2S2 – 9B2 – 4SB + 20S + 30B

 s.t.