 company produces items of type A and type B. For item A the
demand x and price p are related as follows: p = 450 − 2x. For item
B the demand y and price q are related as follows: q = 300 − y.
The cost function is
C(x, y) = x2 + xy + 2y2 + 320x + 220y + 190
(a) Show that the profit is f (x, y) = −3x2 −3y2 −xy+130x+80y−190

To show that the profit function \( f(x, y) \) is given by:

\[
f(x, y) = -3x^2 - 3y^2 - xy + 130x + 80y - 190
\]

we need to express the profit as the difference between the revenue and the cost. Here's how it can be done:

### 1. **Revenue Functions**:
The revenue from selling items \( A \) and \( B \) can be determined by multiplying the price per unit by the quantity sold.

For item \( A \):
\[
\text{Revenue from } A = p \times x = (450 - 2x) \times x = 450x - 2x^2
\]

For item \( B \):
\[
\text{Revenue from } B = q \times y = (300 - y) \times y = 300y - y^2
\]

Therefore, the total revenue \( R(x, y) \) is:
\[
R(x, y) = (450x - 2x^2) + (300y - y^2) = 450x - 2x^2 + 300y - y^2
\]

### 2. **Cost Function**:
The cost function is given by:
\[
C(x, y) = x^2 + xy + 2y^2 + 320x + 220y + 190
\]

### 3. **Profit Function**:
The profit function \( f(x, y) \) is the total revenue minus the total cost:
\[
f(x, y) = R(x, y) - C(x, y)
\]
Substituting the revenue and cost functions:

\[
f(x, y) = (450x - 2x^2 + 300y - y^2) - (x^2 + xy + 2y^2 + 320x + 220y + 190)
\]

### 4. **Simplifying the Profit Function**:

Expand and simplify the expression:

\[
f(x, y) = 450x - 2x^2 + 300y - y^2 - x^2 - xy - 2y^2 - 320x - 220y - 190
\]

Combine like terms:

- The \( x^2 \) terms:
  \[
  -2x^2 - x^2 = -3x^2
  \]

- The \( y^2 \) terms:
  \[
  -y^2 - 2y^2 = -3y^2
  \]

- The \( xy \) term:
  \[
  -xy
  \]

- The \( x \) terms:
  \[
  450x - 320x = 130x
  \]

- The \( y \) terms:
  \[
  300y - 220y = 80y
  \]

- The constant term:
  \[
  -190
  \]

So, the simplified profit function is:

\[
f(x, y) = -3x^2 - 3y^2 - xy + 130x + 80y - 190
\]

This matches the given profit function, thus proving that the profit function is indeed:

\[
f(x, y) = -3x^2 - 3y^2 - xy + 130x + 80y - 190
\]

Would you like any further explanations or have questions?

---

1. How can the profit function be optimized to find the maximum profit?
2. What happens to the profit if the demand for item A increases?
3. How does the interaction term \( xy \) affect the profit?
4. What would the profit function look like if the cost function had different coefficients?
5. How do changes in the price functions \( p \) and \( q \) impact the profit?

**Tip:** In optimization problems, it's essential to understand how changes in variables affect the overall outcome, particularly when dealing with cost and revenue functions.

Learn how to maximize profit by deriving and optimizing cost and revenue functions. This problem involves deriving the profit function from given cost and revenue functions and optimizing it to find maximum profit. Suitable for college-level students.

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