The problem asks which of the given statements provides sufficient additional information to conclude that there is a value $c \in [3, 5]$ such that $f(c) = 4$, given that $f(3) < 4 < f(5)$.

This is a typical application of the Intermediate Value Theorem (IVT), which states that if a function $f$ is continuous on a closed interval $[a, b]$, and $f(a)$ and $f(b)$ take opposite sides of a value $k$, then there exists a value $c \in [a, b]$ such that $f(c) = k$.

Let's evaluate the options:

• A. $f$ is defined for all $x$: This statement does not provide enough information. The function being defined everywhere doesn't guarantee that $f$ takes the value $4$ on the interval [3, 5].

• B. $f$ is increasing for all $x$: While this would be helpful, it doesn't directly provide information about $f(c) = 4$, because the function could be increasing but might still skip over 4. Without knowing continuity, this isn't sufficient.

• C. $f$ is continuous for all $x$: This would be sufficient because the Intermediate Value Theorem applies. Since $f(3) < 4 < f(5)$, the continuity of $f$ guarantees that there is a value $c \in [3, 5]$ such that $f(c) = 4$.

• D. There is a value $c \in [3, 5]$ such that $\lim_{x \to c} f(x) = 4$: This statement discusses a limit, but does not necessarily mean that $f(c) = 4$. The limit of $f(x)$ approaching 4 as $x \to c$ doesn't guarantee the function takes the value $4$ at $c$.

Conclusion:

The correct answer is C, because the continuity of $f$ ensures that the Intermediate Value Theorem applies, and thus there exists a value $c \in [3, 5]$ such that $f(c) = 4$.

Would you like any further clarification on this?

Here are 5 related questions to deepen your understanding:

1. What are the conditions for the Intermediate Value Theorem to apply?
2. Can the Intermediate Value Theorem be applied if the function is discontinuous?
3. How would the problem change if $f(3) > 4$ and $f(5) > 4$?
4. How does the concept of limits differ from function values?
5. What would be an example of a function that does not satisfy the Intermediate Value Theorem?

Tip: When applying the Intermediate Value Theorem, always ensure the function is continuous on the interval you are considering.