The appropriate calculus-based justification for the fact that the function $g$ is increasing is:

Choice C: $g'$ is positive.

Explanation:

• A function $g(x)$ is increasing on an interval if its derivative $g'(x)$ is positive on that interval.
• Since the graph of $g'$ is positive (above the x-axis), this indicates that the slope of $g(x)$ is positive, meaning $g(x)$ is increasing.

Do you want more details or have any questions? Here are some related questions you might find useful:

1. How does the sign of $g'(x)$ affect the concavity of $g(x)$?
2. What does it mean for $g(x)$ if $g'(x)$ is zero at a certain point?
3. How can you determine where $g(x)$ is concave up or concave down using $g''(x)$?
4. What information does the graph of $g'(x)$ give about the critical points of $g(x)$?
5. How can you determine the intervals where $g(x)$ is decreasing?

Tip: To determine whether a function is increasing or decreasing, always look at the sign of its first derivative $g'(x)$. If $g'(x) > 0$, the function is increasing; if $g'(x) < 0$, it is decreasing.