The function f is defined by the given equation, where a, b, and c are distinct constants. When a <
x < b the value of f(x) is positive. The graph of y = f(x) in the xy-plane contains the point (r, s),
where r and s are constants. If s = 8 which of the following could be true?
f(x) = 3(x - a)(x - b)(x - c)
I. r < a
II. b < r < c
III. r > c

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.

The function f is defined by the given equation, where a, b, and c are distinct constants. When a < x < b the value of f(x) is positive. The graph of y = f(x) in the xy-plane contains the point (r, s), where r and s are constants. If s = 8 which of the following could be true? f(x) = 3(x - a)(x - b)(x - c) I. r < a II. b < r < c III. r > c

Explore how to analyze a cubic function f(x) = 3(x - a)(x - b)(x - c) and determine which intervals correspond to positive or negative values. This problem involves understanding root behavior and analyzing intervals for polynomial sign changes. Ideal for Grades 10-12.

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