The question is asking which of the given statements, if true, would be sufficient to conclude that there exists a number $c$ in the interval $[-2, 2]$ such that $g(c) = 6$.

Let's analyze each statement using the Intermediate Value Theorem (IVT), which is often useful when reasoning about whether a function takes a particular value in an interval.

Intermediate Value Theorem (IVT)

The IVT states that if a function $g$ is continuous on a closed interval $[a, b]$, and $g(a) \neq g(b)$, then for every value $y$ between $g(a)$ and $g(b)$, there exists some $c \in (a, b)$ such that $g(c) = y$.

In this problem, we are looking for $c$ such that $g(c) = 6$ in the interval $[-2, 2]$.

From the table, we see that:

• $g(-1) = -1$
• $g(0) = 4$
• $g(1) = 11$

Since $6$ is between $g(0) = 4$ and $g(1) = 11$, if $g$ is continuous on the interval $[0, 1]$, the IVT guarantees that there is a $c \in (0, 1)$ such that $g(c) = 6$.

Now, let's review each statement:

Statement I: "g is defined for all $x$ in the interval $[-2, 2]$."

• This statement alone is not sufficient because simply knowing that $g$ is defined doesn't imply that $g(c) = 6$. We need more information about the behavior of the function (e.g., continuity or increasing/decreasing nature) to apply the IVT.

Statement II: "g is increasing on the interval $[-2, 2]$."

• If $g$ is increasing on the interval $[-2, 2]$, we can conclude that $g$ takes every value between $g(0) = 4$ and $g(1) = 11$. Since $6$ lies between these two values, this statement would be sufficient to conclude that there is a $c \in (0, 1)$ such that $g(c) = 6$.

Statement III: "g is continuous on the interval $[-2, 2]$."

• If $g$ is continuous on $[-2, 2]$, we can apply the Intermediate Value Theorem. Since $g(0) = 4$ and $g(1) = 11$, and 6 is between these values, continuity would guarantee the existence of a $c \in (0, 1)$ such that $g(c) = 6$. Thus, this statement would also be sufficient.

Conclusion:

• Statement II (g is increasing) and Statement III (g is continuous) are individually sufficient to conclude that there exists a number $c$ such that $g(c) = 6$.
• Statement I (g is defined) alone is not sufficient.

Would you like further details or have any questions?

Related Questions:

1. How does the Intermediate Value Theorem apply to non-continuous functions?
2. What if the function was decreasing instead of increasing? How would that affect the conclusion?
3. Can we guarantee $g(c) = 6$ if only the endpoints of the interval are given and $g$ is not continuous?
4. What is the difference between continuity and differentiability in ensuring the existence of solutions like this?
5. Could a function be defined but not continuous on an interval? How does that affect the Intermediate Value Theorem?

Tip: When using the IVT, always check whether the function is continuous over the interval and whether the value you're interested in lies between the function's values at the interval's endpoints.