Problem 333 - Calculus Volume 1

Question

For the following exercises, consider a pizzeria that sell pizzas for a revenue of \(R(x)=a x\) and costs \(C(x)=b+c x+d x^{2},\) where \(x\) represents the number of pizzas. Assume that \(R(x)=10 x\) and \(C(x)=2 x+x^{2}\) . How many pizzas sold maximizes the profit?

Step-by-Step Solution

Answer

Selling 4 pizzas maximizes profit.

Step 1: Define Profit Function

The profit function is defined as the difference between revenue and cost functions. Thus, the profit function can be expressed as: \( P(x) = R(x) - C(x) \) Substitute the given revenue \( R(x) = 10x \) and cost \( C(x) = 2x + x^2 \) into the profit function.\[ \text{P(x)} = (10x) - (2x + x^2) \]

Step 2: Simplify the Profit Function

Simplify the expression for \( P(x) \): \[ P(x) = 10x - 2x - x^2 \] Combine like terms:\[ P(x) = 8x - x^2 \]

Step 3: Find the Derivative of Profit Function

To find the maximum profit, we need to find the critical points by taking the derivative of the profit function and setting it to zero. The derivative of \( P(x) = 8x - x^2 \) is:\[ P'(x) = 8 - 2x \]

Step 4: Solve for Critical Points

Set the derivative equal to zero and solve for \( x \):\[ 8 - 2x = 0 \]Add \( 2x \) to both sides:\[ 8 = 2x \]Divide both sides by 2:\[ x = 4 \]

Step 5: Verify Maximum with Second Derivative

Find the second derivative of the profit function to verify that the critical point \( x = 4 \) is a maximum.The second derivative is:\[ P''(x) = -2 \]Since \( P''(x) = -2 < 0 \), the function is concave down at \( x = 4 \), confirming a local maximum.

Key Concepts

Profit Maximization

In the context of business economics, profit maximization involves finding the number of units that a company must produce and sell in order to achieve the highest possible profit. Profit is simply the total revenue minus total costs. In our pizzeria example, the revenue function is given as \(R(x) = 10x\) and the cost function as \(C(x) = 2x + x^2\). To find the profit function, we subtract the cost function from the revenue function: \[ P(x) = R(x) - C(x) = 10x - (2x + x^2) = 8x - x^2 \]

Profit maximization is achieved when the number of units sold results in the highest difference between revenue and cost. This usually involves calculus because it's about determining at which point this difference attains its maximum value. By establishing a mathematical model like this, the pizzeria can adjust its production accordingly to maximize profit.

Derivative for Critical Points

To find the critical points of a function, we take its derivative and set it equal to zero. This step identifies potential maxima or minima points of the function. For our profit function, \(P(x) = 8x - x^2\), we calculate the derivative: \[ P'(x) = \frac{d}{dx}(8x - x^2) = 8 - 2x \]

After finding the derivative, we solve \(P'(x) = 0\) to find the critical points. In our scenario: \[ 8 - 2x = 0 \] Solving this equation gives: \[ x = 4 \]

At \(x = 4\), there is a critical point. These critical points will tell us where the function could possibly reach its highest or lowest values. These points help businesses decide at what production level the costs and revenues yield the greatest profit.

Second Derivative Test

The second derivative test is a method to verify whether a critical point found via the first derivative is a maximum or minimum. This involves taking the second derivative of the function and evaluating it at the found critical point. For the profit function \(P(x) = 8x - x^2\), the second derivative is determined as: \[ P''(x) = -2 \]

If \(P''(x) < 0\), the function is concave down, suggesting that the critical point is a local maximum. Conversely, if \(P''(x) > 0\), the function is concave up, indicating a local minimum. In our case, the second derivative \(P''(x) = -2\) is negative at \(x = 4\).

• This confirms that the critical point \(x = 4\) is indeed a maximum.
• The pizza restaurant achieves its greatest profit when it sells 4 pizzas.

Using the second derivative test provides an additional level of verification, enabling businesses to confidently proceed with strategies based on their findings.