The question involves the function $f(x) = 3(x - a)(x - b)(x - c)$, where $a$, $b$, and $c$ are distinct constants, and we are asked to analyze some conditions based on a few given statements.

Let's break it down:

1. Form of $f(x)$:
   This is a cubic function with roots at $x = a$, $x = b$, and $x = c$. The sign of $f(x)$ depends on the intervals between these roots, and since $f(x)$ is cubic, it will have turning points and change signs between the roots.

2. Given conditions:

   • The function is positive when $a < x < b$. This indicates that between these two roots, $f(x) > 0$.
   • The graph contains a point $(r, 8)$, which implies that $f(r) = 8$.

3. Analysis of the statements:

   • Statement I: $r < a$.
     If $r < a$, then the function is likely negative since it is a cubic polynomial and has roots at $a$, $b$, and $c$. This contradicts $f(r) = 8$, so this statement is false.

   • Statement II: $b < r < c$.
     Between $b$ and $c$, the function may be positive or negative. However, since $f(r) = 8$, and 8 is positive, this interval is a possible solution. So, this statement could be true.

   • Statement III: $r > c$.
     For $r > c$, the cubic function would tend to positive infinity as $x \to \infty$. Since $f(r) = 8$ is positive, this statement could also be true.

4. Conclusion: The valid options are likely Statements II and III, meaning that the correct answer is (D) II and III only.

Would you like more details or have any further questions? Here are some related questions to consider:

1. What is the general behavior of a cubic function?
2. How do we determine the sign of a polynomial function between its roots?
3. How would the graph of $f(x) = 3(x - a)(x - b)(x - c)$ look near its roots?
4. What are turning points in cubic functions, and how do they affect the function’s behavior?
5. How can we find the exact value of $r$ given the information?

Tip: When analyzing polynomials, always consider the behavior between and beyond the roots to understand where the function is positive or negative.