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The question in the image is asking which statement provides sufficient information to conclude that there is a value \( x = c \) in the interval \([5,7]\) such that \( f(c) = 6 \).

### Let’s break down the problem:

1. We know that \( f(5) < 6 < f(7) \), meaning that \( f(5) \) is less than 6, and \( f(7) \) is greater than 6.
2. The goal is to find whether there exists some \( c \in [5,7] \) such that \( f(c) = 6 \).

### Options:

- **(A)** \( f \) is defined for all \( x \):  
  This isn't sufficient information, as just knowing the function is defined doesn't guarantee anything about its behavior or that the value \( f(c) = 6 \) exists.

- **(B)** \( f \) is increasing for all \( x \):  
  This could be helpful because if \( f \) is increasing and we know \( f(5) < 6 < f(7) \), the Intermediate Value Theorem could be applied. However, this doesn't ensure the continuity of \( f \), which is necessary for the IVT to apply.

- **(C)** \( f \) is continuous for all \( x \):  
  This is more promising. If \( f \) is continuous, and since \( f(5) < 6 < f(7) \), by the **Intermediate Value Theorem** (IVT), there must exist some \( c \in [5, 7] \) such that \( f(c) = 6 \). Continuity guarantees that \( f \) takes all values between \( f(5) \) and \( f(7) \).

- **(D)** There is a value \( x = c \) in the interval \([5,7]\) such that \( \lim_{x \to c} f(x) = 6 \):  
  This is redundant information. The fact that the limit of the function at \( x = c \) is 6 doesn't help directly in proving that there is a point where \( f(c) = 6 \), because the function could approach 6 without actually reaching it.

### Correct Answer: **(C)** \( f \) is continuous for all \( x \).

This option provides sufficient additional information because the **Intermediate Value Theorem** guarantees that a continuous function will take every value between \( f(5) \) and \( f(7) \), including 6, in the interval \([5,7]\).

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Would you like a detailed explanation of the Intermediate Value Theorem or have any other questions?

Here are 5 follow-up questions to extend this concept:
1. How does the Intermediate Value Theorem apply to real-world situations?
2. What are the necessary conditions for the Intermediate Value Theorem to hold?
3. Can the Intermediate Value Theorem be applied if a function is not continuous?
4. How do you prove the existence of a root using the Intermediate Value Theorem?
5. What are some counterexamples where the Intermediate Value Theorem fails?

**Tip**: Continuity of a function is a crucial property when working with limits and solving existence problems like this one. Always check if the function is continuous before applying theorems that depend on it.

Let f be a function such that f(5) < 6 < f(7). Which of the following statements provides sufficient additional information to conclude that there is a value x = c in the interval [5, 7] such that f(c) = 6?

Understand how to apply the Intermediate Value Theorem to find a value x = c in the interval [5, 7] such that f(c) = 6, given that f is continuous and f(5) < 6 < f(7). Learn more about the theorem, continuity, and their roles in solving existence problems.

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