The problem is asking about the point of inflection of the graph of $y = M(d)$, where $M$ is the number of people who visit the attraction, and $R(d)$ represents the rate of change of $M(d)$. The rate $R(d)$ is given as:

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

Key Idea:

1. Point of inflection: A point of inflection occurs where the second derivative $M''(d)$ changes sign. Since $R(d)$ is the first derivative of $M(d)$, the inflection point for $M(d)$ corresponds to where the first derivative $R(d)$ has a critical point or $R'(d) = 0$.
   • To determine this, we analyze $R(d)$.

Steps to Solve:

1. Differentiate $R(d)$ to get $R'(d)$. Set $R'(d) = 0$ and solve for $d$.
2. Critical points of $R(d)$ are candidates for inflection points of $M(d)$.

Given the answer choices, we observe:

• A: $d = 8.627$
• B: $d = 3.894 \text{ and } 13.728$
• C: $d = 5.911 \text{ and } 11.590$
• D: $d = 3.894, 8.627, \text{ and } 13.728$.

Solution:

From the structure of the function, the correct inflection points occur where the derivative $R'(d) = 0$, leading to multiple values of $d$. Through analysis or computational methods, we find:

• The correct answer is (D): $d = 3.894, 8.627, \text{ and } 13.728$.

Key Explanation:

1. $R(d)$ is the derivative of $M(d)$.
2. $R'(d) = 0$ gives the critical points where $M(d)$ has inflection points.
3. By solving $R'(d) = 0$, the values $3.894, 8.627, \text{ and } 13.728$ satisfy this condition.

Let me know if you'd like the detailed steps for differentiation or solving $R'(d) = 0$.

Related Questions:

1. What is the process for finding points of inflection of any function?
2. How do you calculate critical points from a derivative?
3. Why do we analyze $R'(d)$ instead of $R(d)$ for inflection points?
4. How can you verify a point of inflection using sign changes of $M''(d)$?
5. What are practical applications of inflection points in real-world scenarios?

Tip:

To determine inflection points for a function $M(d)$, focus on the second derivative and identify where it changes sign.