Question 1192306: A marketing firm estimate that n months after the introduction of a client of new product f(n) thousand households will use it, where
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! The prompt is incomplete. To find the maximum number of households that will use the product, you need the actual function f(n).
**Here's how you would generally approach this type of problem:**
1. **Find the Derivative:**
* Calculate the derivative of the function f(n) with respect to 'n'. This will give you the rate of change of the number of households using the product over time.
2. **Find Critical Points:**
* Set the derivative equal to zero and solve for 'n'. These values of 'n' represent potential points where the maximum or minimum number of households might occur.
3. **Determine the Maximum:**
* Use the second derivative test or analyze the behavior of the derivative around the critical points to determine which point corresponds to the maximum number of households.
4. **Calculate the Maximum Number of Households:**
* Substitute the value of 'n' that corresponds to the maximum into the original function f(n). This will give you the maximum number of households that will use the product.
**Example:**
Let's say the function is:
f(n) = 100n - n²
* **Find the Derivative:**
* f'(n) = 100 - 2n
* **Find Critical Points:**
* 100 - 2n = 0
* 2n = 100
* n = 50
* **Determine the Maximum:**
* Since the coefficient of the n² term is negative, the parabola opens downwards, indicating a maximum at the critical point.
* **Calculate the Maximum Number of Households:**
* f(50) = 100 * 50 - 50² = 5000 - 2500 = 2500
**Therefore, in this example, the maximum number of households that will use the product is 2500 thousand, or 2,500,000 households.**
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