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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 $$63,000.$ According to Innis' risk

measurement system, each unit invested in the stock fund has a risk index of $19,$ and each unit invested in the money market fund has a risk index of $7$;

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 $$450,000$ be

invested in the money market fund.

$($a$)$ Determine how many units of each fund Innis should purchase for the client to minimize the total risk index for the portfolio. $($Assume $S$ is the

number of units purchased in the stock fund and $M$ is the number of units purchased in the money market fund.$)$

$(S,M)=(,)$

$($b$)$ How much annual income $($in dollars$)$ will this investment strategy generate?

$

$($c$)$ Suppose the client desires to maximize annual return and no longer requires that $$450,000$ be invested in the money market fund. How should the

funds be invested?

Invest half in the stock fund and half in the money market fund.

Invest everything in the stock fund.

Invest everything in the money market fund.