To solve the given problem, let's break it down into steps:

1. Break-even quantity

The break-even point occurs when the revenue equals the cost, i.e., $R(x) = C(x)$. This means we set the two functions equal and solve for $x$:

$R(x) = C(x) \implies -x^2 + 8x = x + 6$

This is the equation that must be solved to find the break-even quantity.

2. Find the minimum break-even quantity

To solve $-x^2 + 8x = x + 6$:

$-x^2 + 8x - x - 6 = 0$ $-x^2 + 7x - 6 = 0$

Multiply through by $-1$ to make the coefficient of $x^2$ positive:

$x^2 - 7x + 6 = 0$

Factor the quadratic equation:

$(x - 6)(x - 1) = 0$

Thus, the solutions are $x = 6$ and $x = 1$.
These are the break-even points, so the minimum break-even quantity is $x = 1$.

3. Find the maximum revenue

The revenue function is $R(x) = -x^2 + 8x$. This is a quadratic function, and since the coefficient of $x^2$ is negative, the parabola opens downward. The maximum revenue occurs at the vertex of the parabola.

The formula for the $x$-coordinate of the vertex for a quadratic equation $ax^2 + bx + c$ is:

$x = \frac{-b}{2a}$

For $R(x) = -x^2 + 8x$, we have $a = -1$ and $b = 8$:

$x = \frac{-8}{2(-1)} = 4$

Thus, the maximum revenue occurs at $x = 4$. To find the value of the maximum revenue, substitute $x = 4$ into $R(x)$:

$R(4) = -(4)^2 + 8(4) = -16 + 32 = 16$

Thus, the maximum revenue is 16 thousand dollars (or $16,000).

4. Find the maximum profit

Profit $P(x)$ is defined as revenue minus cost:

$P(x) = R(x) - C(x)$

Using the given functions $R(x) = -x^2 + 8x$ and $C(x) = x + 6$, we have:

$P(x) = (-x^2 + 8x) - (x + 6)$ $P(x) = -x^2 + 8x - x - 6$ $P(x) = -x^2 + 7x - 6$

This is the expression for the profit function $P(x)$.

5. Find the maximum profit

Since $P(x) = -x^2 + 7x - 6$ is a quadratic function with a negative $x^2$-term, it also has a maximum at its vertex. The $x$-coordinate of the vertex is:

$x = \frac{-7}{2(-1)} = \frac{7}{2} = 3.5$

To find the maximum profit, substitute $x = 3.5$ into $P(x)$:

$P(3.5) = -(3.5)^2 + 7(3.5) - 6$ $P(3.5) = -12.25 + 24.5 - 6 = 6.25$

Thus, the maximum profit is 6.25 thousand dollars (or $6,250).

---

Summary:

• Minimum break-even quantity: $x = 1$
• Maximum revenue: $16,000
• Profit function: $P(x) = -x^2 + 7x - 6$
• Maximum profit: $6,250

---

Would you like more details on any of the steps?

Here are 5 related questions:

1. What does the second break-even quantity represent in this context?
2. How does the vertex formula help in finding maximum or minimum points in quadratic functions?
3. Can the profit function have more than one maximum? Why or why not?
4. What is the economic significance of the break-even point in business?
5. How would increasing the cost function $C(x)$ affect the break-even points?

Tip: Always check whether the quadratic opens upwards or downwards before determining if it has a maximum or minimum value.