Suppose you have [tex]$\$[/tex]16,000[tex]$ to invest in three stocks, A, B, and C. Stock A is a low-risk stock that has expected returns of $[/tex]4\%[tex]$. Stock B is a medium-risk stock that has expected returns of $[/tex]5\%[tex]$. Stock C is a high-risk stock that has expected returns of $[/tex]6\%[tex]$. You want to invest at least $[/tex]\[tex]$1,000$[/tex] in each stock. To balance the risks, you decide to invest no more than [tex]$\$[/tex]7,000[tex]$ in stock C and to limit the amount invested in C to less than 4 times the amount invested in stock A. You also decide to invest less than twice as much in stock B as in stock A. How much should you invest in each stock to maximize your expected profit?

Complete the constraints.

Stock A: $[/tex]x \geq \boxed{1000}[tex]$

Stock B: $[/tex]y \geq \boxed{1000}[tex]$ and $[/tex]y \leq \boxed{2x}[tex]$

Stock C: $[/tex]16,000 - x - y \geq \boxed{1000}[tex]$

$[/tex]16,000 - x - y \leq \boxed{7000}[tex]$

$[/tex]x \geq \boxed{\frac{16,000 - y}{4}}$

Let's denote the investment amounts in stocks A, B, and C as [tex]$x$[/tex], [tex]$y$[/tex], and [tex]$16,000 - x - y$[/tex] respectively since the total amount to be invested is \[tex]$16,000. We need to state the constraints based on the given conditions: 1. Minimum investment in each stock: - Stock A: $x$ - Stock B: $y$ - Stock C: $16,000 - x - y$ Since at least \$[/tex]1,000 must be invested in each stock:
[tex]\[ x \geq 1000 \][/tex]
[tex]\[ y \geq 1000 \][/tex]
[tex]\[ 16000 - x - y \geq 1000 \][/tex]

2. Maximum investment in stock C:
[tex]\[ 16000 - x - y \leq 7000 \][/tex]

3. Limiting stock C to less than 4 times stock A:
[tex]\[ 16000 - x - y \leq 4x \][/tex]

4. Investment in stock B less than twice as much as in stock A:
[tex]\[ y \leq 2x \][/tex]

Now, let's state the constraints clearly:

- Stock A: [tex]$x \geq 1000$[/tex]
- Stock B: [tex]$y \geq 1000$[/tex] and [tex]$y \leq 2x$[/tex]
- Stock C: [tex]\[16000 - x - y \geq 1000\][/tex]

[tex]\[ 16000 - x - y \leq 7000 \][/tex]

[tex]\[ 16000 - x - y \leq 4x \][/tex]

Thus, the complete constraints are:

- Stock A: [tex]$x \geq 1000$[/tex]
- Stock B: [tex]$y \geq 1000$[/tex] and [tex]$y \leq 2x$[/tex]
- Stock C: [tex]\[16000 - x - y \geq 1000\][/tex]

[tex]\[ 16000 - x - y \leq 7000 \][/tex]

[tex]\[ 16000 - x - y \leq 4x \][/tex]