2 A factory produces smartphones, where the daily output Pix) (in units) depends on the number of employees x. The production is modeled by the function:

P(x)=-2x20x60x70x

a) Use ICT to draw the graph of P(x).

b) Determine the practical domain and practical range.

c) Determine the zeros. What do the zeros mean in this context?

d) Draw the value progression.

e) Use the graph and the characteristics you found to determine how many workers should be employed to maximize production.

### Graph Analysis
The graph of \( P(x) = -2x(x-20)(x-60)(x-70) \) shows a quartic function with the following key features:
- The function has **zeros** at \( x = 0 \), \( x = 20 \), \( x = 60 \), and \( x = 70 \).
- The production peaks between the zeros, and the function behaves as expected: opening downwards due to the negative leading coefficient.

Now let's continue with the analysis based on the graph:

### b) Practical Domain and Practical Range
- **Practical Domain**: \( [0, 70] \), since the number of employees must be non-negative and up to 70.
- **Practical Range**: From the graph, the maximum production value lies somewhere between \( 20 \leq x \leq 60 \). We will calculate this maximum production value next to determine the practical range.

### c) Zeros of \( P(x) \)
The zeros occur at \( x = 0 \), \( x = 20 \), \( x = 60 \), and \( x = 70 \), meaning no production occurs when the factory employs 0, 20, 60, or 70 workers.

### d) Value Progression
The value progression shows that the production starts at zero, increases to a maximum somewhere between 20 and 60 employees, and then decreases back to zero at 70 employees.

### e) Maximizing Production
Next, we'll determine the number of employees needed to maximize production by finding the critical points between the zeros. I'll calculate the maximum value of \( P(x) \).### Maximum Production
The production is maximized when approximately **8 employees** are employed, resulting in a maximum production output of about **619,544 units**.

### Summary:
- **Practical Domain**: \( [0, 70] \)
- **Practical Range**: \( [0, 619,544] \)
- **Zeros**: \( x = 0 \), \( x = 20 \), \( x = 60 \), \( x = 70 \), indicating no production occurs at these employee levels.
- **Maximum Production**: The production reaches its peak when approximately 8 employees are hired.

Would you like further details on any of these steps or additional clarification?

Here are 5 related questions:
1. How would the production model change if more zeros were added?
2. What is the significance of the negative coefficient in \( P(x) \)?
3. How could this model be adjusted for different workforce sizes?
4. What does the shape of the graph tell us about the efficiency of the factory?
5. Can we predict production trends outside of the 70-employee limit?

**Tip**: Always check for the practical domain and range in real-world problems to avoid unrealistic results.

A factory produces smartphones, where the daily output P(x) (in units) depends on the number of employees x. The production is modeled by the function: P(x) = -2x(x-20)(x-60)(x-70). a) Use ICT to draw the graph of P(x). b) Determine the practical domain and practical range. c) Determine the zeros. What do the zeros mean in this context? d) Draw the value progression. e) Use the graph and the characteristics you found to determine how many workers should be employed to maximize production.

Explore how to maximize smartphone production in a factory using the quartic function P(x) = -2x(x-20)(x-60)(x-70). This problem covers polynomial graph analysis, zeros, and production optimization. Learn to determine practical domain, range, and critical points. Suitable for students in Grades 11-12.

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