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The problem involves using the Intermediate Value Theorem (IVT) to determine the minimum number of zeros of the function \( f(x) \) on the interval \([-6, -2]\).

### **Restating the Problem:**
You are given a table with values of \( f(x) \) for certain values of \( x \) on the interval \([-6, -2]\):

\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-6 & 0 \\
-5 & 3 \\
-4 & 2 \\
-3 & -1 \\
-2 & 2 \\
\hline
\end{array}
\]

You need to determine the minimum number of zeros of \( f(x) \) guaranteed by the Intermediate Value Theorem.

### **Applying the Intermediate Value Theorem (IVT):**
The Intermediate Value Theorem states that if a function \( f(x) \) is continuous on a closed interval \([a, b]\) and \( f(a) \) and \( f(b) \) have opposite signs, then there exists at least one value \( c \) in the interval \((a, b)\) such that \( f(c) = 0 \).

1. **From \( x = -6 \) to \( x = -5 \)**: 
   - \( f(-6) = 0 \) and \( f(-5) = 3 \). 
   - Since \( f(-6) = 0 \), there is at least one zero between \([-6, -5]\).

2. **From \( x = -5 \) to \( x = -4 \)**:
   - \( f(-5) = 3 \) and \( f(-4) = 2 \).
   - Both values are positive, so no zero is guaranteed here.

3. **From \( x = -4 \) to \( x = -3 \)**:
   - \( f(-4) = 2 \) and \( f(-3) = -1 \).
   - The function changes from positive to negative, so there is at least one zero between \([-4, -3]\).

4. **From \( x = -3 \) to \( x = -2 \)**:
   - \( f(-3) = -1 \) and \( f(-2) = 2 \).
   - The function changes from negative to positive, so there is at least one zero between \([-3, -2]\).

### **Conclusion:**
By analyzing the sign changes between consecutive values of \( f(x) \), there are at least **three zeros** of \( f(x) \) on the interval \([-6, -2]\), as guaranteed by the Intermediate Value Theorem.

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Do you want more details, or do you have any questions?

Here are five related questions for further exploration:
1. How would the problem change if the function \( f(x) \) were not continuous?
2. Can you identify any specific intervals where more than one zero might occur?
3. How does the IVT apply if there were more points given in the table?
4. How could you use the IVT to find the approximate location of these zeros?
5. What are some real-world applications of the Intermediate Value Theorem?

**Tip:** Always check for sign changes between function values to easily identify intervals where the IVT guarantees the existence of zeros.

The function f(x) is continuous on the interval [-6, -2]. The table below gives some of its values. What is the minimum number of zeros that f(x) is guaranteed to have in the interval [-6, -2] by the Intermediate Value Theorem?

Learn how to determine the minimum number of zeros of a function on the interval [-6, -2] using the Intermediate Value Theorem. This problem explores key calculus concepts, including continuity and the application of IVT to find zeros.

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