\begin{tabular}{l} (1 point) A company has found that the relationship between the price $$ p $$ and the demand $$ x $$ for a particular product is given approximately by \ The company also knows that the cost of producing the product is given by $$ C(x)=870+399 x $$. \ Find $$ P(x) $$, the profit function. \ $$ P(x)= $$ \ Now use the profit function to do the following: \ (A) Find the average of the $$ x $$ values of all local maxima of $$ P $$. \ Note: If there are no local maxima, enter -1000 . \ Average of $$ x $$ values $$ = $$ \ (B) Find the average of the $$ x $$ values of all local minima of $$ P $$. \ Note: If there are no local minima, enter -1000 . \ Average of $$ x $$ values = $$ \square $$ \ (C) Use interval notation to indicate where $$ P(x) $$ is concave up. \ Note: Enter 'I' for $$ \infty $$, '-l' for $$ -\infty $$, and 'U' for the union symbol. \ If you have extra boxes, fill each in with an ' $$ x $$ '. \ Concave up: \ (D) Use interval notation to indicate where $$ P(x) $$ is concave down. \ Concave down: \ T: \ \hline\end{tabular}

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UpStudy AI · Accepted Answer

To find the profit function $$ P(x) $$, we need to use the relationship between price $$ p $$ and demand $$ x $$, along with the cost function $$ C(x) = 870 + 399x $$. The profit function is defined as:

$$
P(x) = R(x) - C(x)
$$

where $$ R(x) $$ is the revenue function, which can be expressed as:

$$
R(x) = p \cdot x
$$

### Step 1: Define the Revenue Function $$ R(x) $$

Assuming the relationship between price $$ p $$ and demand $$ x $$ is given by a function (which is not provided in the prompt), we can express $$ R(x) $$ as:

$$
R(x) = p(x) \cdot x
$$

### Step 2: Define the Profit Function $$ P(x) $$

Now, substituting $$ R(x) $$ and $$ C(x) $$ into the profit function:

$$
P(x) = R(x) - C(x) = p(x) \cdot x - (870 + 399x)
$$

### Step 3: Find Local Maxima and Minima

To find the local maxima and minima of $$ P(x) $$, we need to take the derivative $$ P'(x) $$ and set it to zero:

$$
P'(x) = R'(x) - C'(x)
$$

Where $$ C'(x) = 399 $$. We will also need to find the second derivative $$ P''(x) $$ to determine concavity.

### Step 4: Solve for Local Maxima and Minima

1. Set $$ P'(x) = 0 $$ to find critical points.
2. Use the second derivative test to classify these points as local maxima or minima.

### Step 5: Calculate Averages and Intervals

- For part (A), find the average of the $$ x $$ values of all local maxima.
- For part (B), find the average of the $$ x $$ values of all local minima.
- For part (C), determine the intervals where $$ P(x) $$ is concave up.
- For part (D), determine the intervals where $$ P(x) $$ is concave down.

Since the specific function for $$ p(x) $$ is not provided, I cannot compute the exact values. If you provide the function $$ p(x) $$, I can proceed with the calculations.
To find the profit function $$ P(x) $$, subtract the cost from the revenue: $$ P(x) = R(x) - C(x) = p(x) \cdot x - (870 + 399x) $$ To find local maxima and minima, take the derivative of $$ P(x) $$, set it to zero, and use the second derivative to classify the points. Then, calculate the averages of the $$ x $$ values for maxima and minima, and determine the intervals where $$ P(x) $$ is concave up or down. However, without the specific function for $$ p(x) $$, I cannot provide numerical answers.

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Solución

Step 1: Define the Revenue Function

Step 2: Define the Profit Function

Step 3: Find Local Maxima and Minima

Step 4: Solve for Local Maxima and Minima

Step 5: Calculate Averages and Intervals

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