Chapter 2.3, Problem 45E

Index-Fund Fees When a mutual fund company charges a fee of $0.47\%$ on its index funds, its assets in the fund are $\$41$ billion. And when it charges a fee of $0.18\%$, its assets in the fund are $\$300$ billion. (Source: The Boston Globe.)

Let $x\%$ denote the fee that the company charges the index fund and $A(x)$ its assets in the fund. Express $A(x)$ as a linear function of $x.$

In September $2004$, Fidelity Mutual lowered its fees on various index funds from an average of $0.3\%$ to $0.10\%$. Let $R(x)$ denote the revenue of the company from fees when its index-fund fee is $x\%$. Compare the revenue of the company before and after lowering the fees. [Hint: Revenue is $x°/0$ of assets.]

Find the fee that maximizes the revenue of the company and determine the maximum revenue.