Question 1198785: Given the profit function π(Q) = -4Q^3 +237Q^2 - 1314Q-8000 where π is in dollar and Q is in ton
A. Find and characterize the critical value(s) of the function
B. With the restriction that output is non-negative, is the profit function concave over its entire domain? Is it convex over its entire domain?
C. If your answer to b above is “No” in both cases, find out the interval over which it is concave, the interval over which it is convex, and comment on the dividing line. If your answer to b is different, come up with your own function which has both concave and convex parts and describe the situation.
Answer by textot(100)   (Show Source):
You can put this solution on YOUR website! ### 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!
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