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 $[5,7]$.

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.