Let's break down the information provided from the image to answer the true, false, or maybe questions:

Given Information:

• The function $g(x)$ is continuous for all real numbers $x$.
• The table provides the sign of the derivative $g'(x)$ over different intervals of $x$.

The key idea to remember when looking at critical numbers (or critical points) is that these are values of $x$ where the derivative is either zero or undefined. Additionally, if a derivative changes sign at a critical point, that point is a local maximum, local minimum, or saddle point.

Now let's analyze each of the statements:

1. $-5$ is a critical number (critical point).

   • From the table, at $x = -5$, $g'(x)$ is undefined. Since the derivative is undefined at $x = -5$, this is a critical point.

   Answer: TRUE.

2. $1$ is a critical number (critical point).

   • At $x = 1$, $g'(x)$ is positive for $x > 1$ and negative for $x < 1$, indicating a sign change around $x = 1$. Therefore, $x = 1$ is a critical point.

   Answer: TRUE.

3. $g(1)$ is the absolute minimum of $g$.

   • From the table, the derivative $g'(x)$ changes from negative to positive at $x = 1$. This suggests that $x = 1$ is a local minimum, but without further context (such as the behavior of the function at other values of $x$), we cannot conclude whether this is the absolute minimum.

   Answer: MAYBE.

Would you like to go over further details of this problem or discuss any of the concepts in more depth?

Here are some related questions for further practice:

1. What are the conditions for a point to be a local maximum or minimum?
2. How do you identify critical points from the derivative of a function?
3. Can a critical point occur where the derivative does not exist?
4. What does it mean for a function to have an absolute minimum or maximum?
5. How can you determine if a critical point is a saddle point?

Tip: Always check if the derivative changes sign at critical points to determine whether the point is a local minimum or maximum!