Question 1198785
### Part A: Critical Values of the Profit Function

The profit function is:
\[
\pi(Q) = -4Q^3 + 237Q^2 - 1314Q - 8000
\]

#### Step 1: First Derivative
The critical points are found by solving \( \pi'(Q) = 0 \). Compute the first derivative:
\[
\pi'(Q) = -12Q^2 + 474Q - 1314
\]

Set \( \pi'(Q) = 0 \):
\[
-12Q^2 + 474Q - 1314 = 0
\]
Divide through by \(-6\) for simplicity:
\[
2Q^2 - 79Q + 219 = 0
\]

Solve using the quadratic formula:
\[
Q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \text{ where } a = 2, b = -79, c = 219.
\]
Substitute:
\[
Q = \frac{-(-79) \pm \sqrt{(-79)^2 - 4(2)(219)}}{2(2)}
\]
\[
Q = \frac{79 \pm \sqrt{6241 - 1752}}{4}
\]
\[
Q = \frac{79 \pm \sqrt{4489}}{4}
\]
\[
Q = \frac{79 \pm 67}{4}
\]

Thus:
\[
Q_1 = \frac{79 + 67}{4} = \frac{146}{4} = 36.5, \quad Q_2 = \frac{79 - 67}{4} = \frac{12}{4} = 3.
\]

Critical values: \( Q_1 = 36.5 \), \( Q_2 = 3 \).

---

#### Step 2: Characterize Critical Points
To determine whether the critical points are maxima, minima, or points of inflection, evaluate the second derivative:
\[
\pi''(Q) = -24Q + 474
\]

At \( Q_1 = 36.5 \):
\[
\pi''(36.5) = -24(36.5) + 474 = -876 + 474 = -402 \quad (\text{negative, local maximum}).
\]

At \( Q_2 = 3 \):
\[
\pi''(3) = -24(3) + 474 = -72 + 474 = 402 \quad (\text{positive, local minimum}).
\]

---

### Part B: Concavity and Convexity Over the Entire Domain
The concavity of the function is determined by the sign of \( \pi''(Q) \):
\[
\pi''(Q) = -24Q + 474
\]

1. **When \( \pi''(Q) < 0 \):**
   \[
   -24Q + 474 < 0 \implies Q > \frac{474}{24} = 19.75
   \]
   The function is **concave** for \( Q > 19.75 \).

2. **When \( \pi''(Q) > 0 \):**
   \[
   -24Q + 474 > 0 \implies Q < 19.75
   \]
   The function is **convex** for \( Q < 19.75 \).

---

### Part C: Intervals of Concavity and Convexity

- The function is **convex** for \( Q \in [0, 19.75) \).
- The function is **concave** for \( Q \in (19.75, \infty) \).

**Dividing Line**: \( Q = 19.75 \) is the point where the second derivative changes sign. This is the **inflection point**, where the function transitions from convexity to concavity.

---

### Comments

- The dividing line (\( Q = 19.75 \)) represents the inflection point, where the slope changes behavior. For \( Q < 19.75 \), the profit function grows at an increasing rate, while for \( Q > 19.75 \), the profit function grows at a decreasing rate.

Let me know if you'd like further clarification or analysis!