A large corporation with monopolistic control in the marketplace has its average daily costs, in dollars, given by
C =
700
x
+ 100x + x2.
The daily demand for x units of its product is given by
p = 360,000 − 50x dollars.
Find the quantity that gives maximum profit.
x =
3521.57

Incorrect: Your answer is incorrect.
units

Find the maximum profit.
$
635,765,609

Incorrect: Your answer is incorrect.

What selling price should the corporation set for its product?

Let's solve the problem step by step.

### Step 1: Define the Functions
Given:
- The cost function $ C(x) = \frac{700}{x} + 100x + x^2 $.
- The demand function $ p(x) = 360,000 - 50x $.

### Step 2: Revenue Function
The revenue function $ R(x) $ is given by the product of the price per unit $ p(x) $ and the quantity sold $ x $:
\[
R(x) = p(x) \times x = (360,000 - 50x) \times x = 360,000x - 50x^2
\]

### Step 3: Profit Function
The profit function $ P(x) $ is the difference between the revenue and cost:
\[
P(x) = R(x) - C(x) = (360,000x - 50x^2) - \left(\frac{700}{x} + 100x + x^2\right)
\]
Simplifying the profit function:
\[
P(x) = 360,000x - 50x^2 - \frac{700}{x} - 100x - x^2
\]
\[
P(x) = 360,000x - 50x^2 - 100x - x^2 - \frac{700}{x}
\]
\[
P(x) = 360,000x - 51x^2 - 100x - \frac{700}{x}
\]
\[
P(x) = 359,900x - 51x^2 - \frac{700}{x}
\]

### Step 4: Find the Critical Points
To find the quantity $ x $ that gives maximum profit, we take the derivative of the profit function with respect to $ x $ and set it to zero.

\[
\frac{dP}{dx} = 359,900 - 102x + \frac{700}{x^2}
\]
Setting $ \frac{dP}{dx} = 0 $ to find the critical points:
\[
359,900 - 102x + \frac{700}{x^2} = 0
\]

This equation is nonlinear and difficult to solve analytically, so it would typically require numerical methods or a calculator to solve for $ x $.

### Step 5: Calculate the Selling Price
The selling price per unit is given by the demand function $ p(x) $:
\[
p(x) = 360,000 - 50x
\]

After determining the optimal value of $ x $ from the derivative equation, you would substitute it back into $ p(x) $ to find the selling price.

### Summary:
1. **Correct the critical point calculation**: Solve $ 359,900 - 102x + \frac{700}{x^2} = 0 $ for $ x $.
2. **Find the selling price**: Substitute the correct $ x $ value into $ p(x) = 360,000 - 50x $.

Would you like help solving the derivative equation or have any questions?

Learn how to maximize profit and set selling price for a corporation using advanced mathematical functions. Solve the problem step-by-step with detailed formulas and calculations. Suitable for advanced mathematics students and professionals.

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