And new special attraction opened at a museum. The museum management tracked the number of people who visited the attraction each day created a function model M for the number of people for each day d after the attraction opened. each day they also calculated the rate of change of the number of people visiting your attraction. They created a function model R for the rate of change people per day each day d after the attraction opened. The function is given by R(d)=1/200(-d^4+35d^3-411d^2+1845d-2686.5) which of the following values of D does graph of Y=m(d) point of inflection? A.8.627 only B. 3.894 and 13.728 only. C. 5.911 and 11.590. D. 3.894,8.627 and 13.728.

To find the point of inflection of the graph of \( y = M(d) \), we need to analyze the given rate of change function, \( R(d) \). The function \( R(d) \) represents the derivative of \( M(d) \), i.e., \( R(d) = M'(d) \).

A point of inflection of the graph of \( M(d) \) occurs where its second derivative, \( M''(d) \), changes sign. Since \( R(d) = M'(d) \), we find the second derivative \( M''(d) \) by differentiating \( R(d) \).

The function for the rate of change is given by:
\[
R(d) = \frac{1}{200} \left( -d^4 + 35d^3 - 411d^2 + 1845d - 2686.5 \right)
\]

### Step 1: Compute \( R'(d) = M''(d) \)

To find the inflection points, we need to find the derivative of \( R(d) \) and set it to zero:

\[
R'(d) = \frac{d}{dd} \left( \frac{1}{200} \left( -d^4 + 35d^3 - 411d^2 + 1845d - 2686.5 \right) \right)
\]

This simplifies to:

\[
R'(d) = \frac{1}{200} \left( \frac{d}{dd} \left( -d^4 + 35d^3 - 411d^2 + 1845d - 2686.5 \right) \right)
\]

Now, differentiate the polynomial:

\[
R'(d) = \frac{1}{200} \left( -4d^3 + 105d^2 - 822d + 1845 \right)
\]

### Step 2: Set \( R'(d) = 0 \) to Find Critical Points

To find the inflection points, we solve:

\[
-4d^3 + 105d^2 - 822d + 1845 = 0
\]

Let me solve this cubic equation to find the values of \( d \).The cubic equation produced complex expressions for the roots. To get the numerical approximations for the critical points, I'll compute them to determine which of the given options match.The numerical approximations of the critical points are:

1. \( d \approx 8.627 \)
2. \( d \approx 3.894 \)
3. \( d \approx 13.728 \)

These values match the options provided. The correct answer is:

**D. 3.894, 8.627, and 13.728.**

These are the points where the graph of \( y = M(d) \) has points of inflection.

Would you like a detailed explanation of how these points relate to the concavity of the graph? Here are some related questions for further exploration:

1. What does a point of inflection indicate about a function's graph?
2. How do we determine the concavity of a function using the second derivative?
3. What is the significance of the rate of change function in this context?
4. How do cubic equations generally behave and how can they be solved analytically?
5. What methods can be used to find roots of higher-degree polynomials?

**Tip:** Understanding points of inflection helps in analyzing the overall shape and behavior of a graph, especially in modeling real-world scenarios like visitor trends in this case.

And new special attraction opened at a museum. The museum management tracked the number of people who visited the attraction each day and created a function model M for the number of people for each day d after the attraction opened. Each day they also calculated the rate of change of the number of people visiting the attraction. They created a function model R for the rate of change of people per day each day d after the attraction opened. The function is given by R(d) = 1/200(-d^4 + 35d^3 - 411d^2 + 1845d - 2686.5). Which of the following values of d does the graph of Y=M(d) have a point of inflection? A.8.627 only B. 3.894 and 13.728 only. C. 5.911 and 11.590. D. 3.894, 8.627, and 13.728.

Learn how to find the inflection points of a polynomial function given its rate of change model. This problem involves advanced calculus concepts, including the second derivative test and inflection points. Ideal for students in Grades 11-12.

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