We are given the function $f(x) = 3(x - a)(x - b)(x - c)$, where $a$, $b$, and $c$ are distinct constants. Additionally, it's stated that $f(x) > 0$ for $a < x < b$, and the graph of $f(x)$ contains the point $(r, s)$, with $s = 8$.

Our goal is to determine which of the statements I, II, and III could be true about $r$ in relation to $a$, $b$, and $c$.

Analyzing the function:

The function $f(x) = 3(x - a)(x - b)(x - c)$ is a cubic polynomial with roots at $x = a$, $x = b$, and $x = c$. These roots cause the function to change signs across different intervals:

• When $x < a$, the function will be negative (as all factors are negative or positive depending on the order of the roots).
• In the interval $a < x < b$, the function is positive (since two of the factors are negative, making the product positive).
• When $b < x < c$, the function is negative again.
• For $x > c$, the function becomes positive once more.

Thus, the intervals where $f(x) > 0$ are $x \in (a, b)$ and $x > c$, while $f(x) < 0$ for $x \in (b, c)$ and $x < a$.

Given information:

• The graph contains the point $(r, s)$, and $s = 8$, meaning $f(r) = 8$.
• Since $f(r) = 8$ is positive, $r$ must lie in an interval where the function is positive. From the analysis of $f(x)$, the two intervals where the function is positive are:
  1. $r \in (a, b)$
  2. $r > c$

Evaluating the statements:

• Statement I: $r < a$
  In this case, $f(x)$ is negative for $x < a$, so $f(r) = 8$ cannot occur in this region. Thus, Statement I is false.

• Statement II: $b < r < c$
  In this interval, $f(x)$ is negative, so $f(r) = 8$ cannot occur here either. Therefore, Statement II is false.

• Statement III: $r > c$
  In this region, $f(x) > 0$, so it is possible that $f(r) = 8$. Hence, Statement III is true.

Conclusion:

Only Statement III could be true.

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Would you like any further details or explanations on this?

Here are 5 related questions you might find helpful:

1. How can we find the turning points of a cubic function?
2. How does the behavior of the function change at each root?
3. What is the significance of the leading coefficient in determining the end behavior of a polynomial?
4. How do we analyze the intervals of positivity and negativity for polynomials?
5. What methods can be used to determine whether a given point lies on the graph of a function?

Tip: When analyzing polynomials, always consider the sign changes at each root to determine where the function is positive or negative.