Use the function shown below to model the revenue​ (in millions of​ dollars) from the cell phone industry in a particular​ country, where xequals=2 corresponds to the year 2002. According to this​ model, in what year does revenue reach its​ maximum?
​f(x)equals=54.46 left parenthesis 49.4 x minus 142.3 right parenthesis e Superscript negative 1.5 x54.46(49.4x−142.3)e−1.5x
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Part 1
The revenue reaches its maximum in the year enter your response here.
​(Type a whole​ number.)

To find the year when the revenue reaches its maximum, we need to find the maximum value of the function \( f(x) = 54.46 (49.4x - 142.3) e^{-1.5x} \). This involves finding the critical points by taking the derivative of \( f(x) \) with respect to \( x \), setting it to zero, and solving for \( x \). Then, we can verify if the critical point corresponds to a maximum.

Let's go through the steps:

1. **Differentiate the function \( f(x) \)**.
2. **Set the derivative equal to zero to find the critical points**.
3. **Determine if the critical points correspond to a maximum by using the second derivative test or analyzing the sign of the first derivative**.

First, we compute the derivative of the function.

Given:
\[ f(x) = 54.46 (49.4x - 142.3) e^{-1.5x} \]

Let's differentiate this function with respect to \( x \).The derivative of the function \( f(x) \) is:

\[ f'(x) = -1.5(2690.324x - 7749.658)e^{-1.5x} + 2690.324e^{-1.5x} \]

Setting \( f'(x) = 0 \), we find a critical point at \( x \approx 3.547 \).

Now, to determine the corresponding year, we use the relationship \( x = 2 \) corresponds to the year 2002. Thus:

\[
\text{Year} = 2002 + (x - 2)
\]

Substituting the value of \( x \):

\[
\text{Year} = 2002 + (3.547 - 2) \approx 2003.547
\]

Since the result should be a whole number, we round to the nearest year, which gives **2004** as the year when the revenue reaches its maximum.

Would you like a detailed explanation of the steps or have any further questions?

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### Relative Questions:
1. How do you determine the critical points of a function?
2. What does the first derivative tell us about a function?
3. Why do we set the derivative equal to zero to find maximum points?
4. How can the second derivative test help in identifying maximum points?
5. What is the significance of the coefficient \( e^{-1.5x} \) in the function?
6. How does the base year relate to the variable \( x \) in the function?
7. Can you explain the process of rounding when dealing with real-world data?
8. What are some other methods to find the maximum of a function if derivatives are complex?

### Tip:
When finding the maximum of a function, always check both the critical points and the endpoints of the domain if it's bounded. This ensures that no maximum point is overlooked.

Learn how to find the year when revenue reaches its maximum using a mathematical model. Explore the calculus of derivatives and exponential functions applied to real-world data from the cell phone industry. Suitable for advanced high school students and those interested in applied mathematics.

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