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We are given that \( f(x) \) is a continuous function on the interval \([-1, 3]\), with the following values:

\[
f(-1) = 2, \quad f(0) = 6, \quad f(1) = -5, \quad f(2) = 1, \quad f(3) = -6
\]

We need to determine which of the statements is always true given these properties. Let's analyze each one:

### Statement A:
"The equation \( f(x) = 0 \) has no solutions for \( x \) in the interval \((-1, 0)\)."

- \( f(-1) = 2 \) and \( f(0) = 6 \), so \( f(x) \) is positive on the entire interval \([-1, 0]\) and does not cross zero.
- This means there are no solutions to \( f(x) = 0 \) in the interval \((-1, 0)\), so **this statement is true**.

### Statement B:
"The equation \( f(x) = 0 \) has exactly one solution for \( x \) in the interval \((1, 2)\)."

- \( f(1) = -5 \) and \( f(2) = 1 \), so by the Intermediate Value Theorem (since \( f(x) \) is continuous), \( f(x) \) must cross zero exactly once between 1 and 2.
- **This statement is true**.

### Statement C:
"The equation \( f(x) = 0 \) has at least one solution for \( x \) in the interval \((-1, 0)\)."

- We already know from analyzing Statement A that there are no solutions for \( f(x) = 0 \) in \((-1, 0)\), so **this statement is false**.

### Statement D:
"The equation \( f(x) = 0 \) has at least two solutions for \( x \) in the interval \((1, 3)\)."

- \( f(1) = -5 \), \( f(2) = 1 \), and \( f(3) = -6 \). By the Intermediate Value Theorem, \( f(x) \) must cross zero once between 1 and 2, and again between 2 and 3.
- Thus, there are two solutions in \((1, 3)\), so **this statement is true**.

### Statement E:
"The equation \( f(x) = 0 \) has exactly two solutions for \( x \) in the interval \((1, 3)\)."

- From the reasoning in Statement D, we know that \( f(x) = 0 \) crosses exactly once between 1 and 2, and again between 2 and 3, so there are exactly two solutions in this interval.
- **This statement is true**.

### Conclusion:

The statements **A**, **B**, **D**, and **E** are true, but the question asks for which statement is **always** true. Based on the options presented in the multiple-choice format, the correct answer is:

**E**. \( f(x) = 0 \) has exactly two solutions in the interval \( (1, 3) \).

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Would you like a more detailed explanation of the Intermediate Value Theorem, or do you have any other questions about the problem? Here are some related questions you might find helpful:

1. What is the Intermediate Value Theorem, and how is it used in this problem?
2. How do you determine the number of solutions to an equation involving a continuous function?
3. Could there ever be more than two solutions for \( f(x) = 0 \) in the interval \( (1, 3) \)?
4. How does continuity affect the behavior of functions and their roots?
5. How would the analysis change if \( f(x) \) were not continuous?

**Tip:** The Intermediate Value Theorem is a powerful tool when dealing with continuous functions and can help in determining when a function must cross a certain value, such as zero.

Suppose that f(x) is a continuous function on the closed interval [-1, 3] and that f(-1)=2, f(0)=6, f(1)=-5, f(2)=1, f(3)=-6. Which of the following statements is always TRUE for a function with these properties?

Analyze a continuous function f(x) on the interval [-1, 3] and use the Intermediate Value Theorem to determine the number of solutions to f(x)=0 in specific subintervals. This problem covers key calculus concepts and theorem applications suitable for students in Grades 11-12.

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