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 $(1, 3)$.

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.