A firm has the following total cost and demand function: `C(x)=x^3/3-7x^2+11x+50,x=100-p` Find the profit function.

Answer

To find the profit function given the total cost and demand functions, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Cost Function and Demand Function**:
- The total cost function is given by:
\[ C(x) = \frac{x^3}{3} - 7x^2 + 11x + 50 \]
- The demand function is given by:
\[ x = 100 - p \]
- From the demand function, we can express price \( p \) in terms of \( x \):
\[ p = 100 - x \]

2. **Determine the Revenue Function**:
- The revenue function \( R(x) \) is defined as the product of price and quantity:
\[ R(x) = p \cdot x \]
- Substituting \( p \) from the demand function:
\[ R(x) = (100 - x) \cdot x = 100x - x^2 \]

3. **Calculate the Profit Function**:
- The profit function \( \Pi(x) \) is defined as revenue minus cost:
\[ \Pi(x) = R(x) - C(x) \]
- Substituting the expressions for \( R(x) \) and \( C(x) \):
\[ \Pi(x) = (100x - x^2) - \left(\frac{x^3}{3} - 7x^2 + 11x + 50\right) \]

4. **Simplify the Profit Function**:
- Distributing the negative sign and combining like terms:
\[ \Pi(x) = 100x - x^2 - \frac{x^3}{3} + 7x^2 - 11x - 50 \]
- Combine the terms:
\[ \Pi(x) = -\frac{x^3}{3} + (100x - 11x) + (-x^2 + 7x^2) - 50 \]
- This simplifies to:
\[ \Pi(x) = -\frac{x^3}{3} + 89x + 6x^2 - 50 \]

5. **Final Profit Function**:
- Thus, the profit function is:
\[ \Pi(x) = -\frac{x^3}{3} + 6x^2 + 89x - 50 \]