Innis Investments manages funds for a number of companies and wealthy clients. The investment strategy is tailored to each client's needs.

For a new client, Innis has been authorized to invest up to $$1\mathrm{.}2$ million in two investment funds: a stock fund and a money market fund. Each unit of the stock fund costs $$50$ and provides an annual

rate of return of $10\%$; each unit of the money market fund costs $$100$ and provides an annual rate of return of $4\%.$

The client wants to minimize risk subject to the requirement that the annual income from the investment be at least $$60,000.$

According to Innis' risk measurement system, each unit invested in the stock fund has a risk index of $8,$ and each unit invested in the money market fund has a risk index of $3.$ The higher risk index

associated with the stock fund simply indicates that it is the riskier investment.

Innis's client also specified that at least $$300,000$ be invested in the money market fund.

Refer to the computer solution shown below.

$($a$)$ Suppose the risk index for the stock fund $($the value of $C_S)$ increases from its current value of $8$ to $16.$ How does the optimal solution change, if at all?

The optimal solution becomes $(9600,3000).$

The optimal solution does not change.

$($b$)$ Suppose the risk index for the money market fund $($the value of $C_M)$ increases from its current value of $3$ to $5\mathrm{.}5.$ How does the optimal solution change, if at all?

The optimal solution becomes $(9600,3000).$

The optimal solution does not change.

$($c$)$ Suppose $C_S$ increases to $16$ and $C_M$ increases to $5\mathrm{.}5.$ How does the optimal solution change, if at all?

The optimal solution becomes $(9600,3000).$

The optimal solution does not change.