The differentiable function 
\[g\] and its derivative 
\[g'\] are graphed.
Functions g and g prime are graphed. The x-axis is unnumbered. Each graph is a curve. The graph of function g starts in quadrant 4, moves upward concave down, and approaches the x-axis. The graph of function g prime starts in quadrant 1, moves downward concave up, and approaches the x-axis.
\[y\]
\[x\]
\[\blueD{g}\]
\[\goldD{g'}\]
What is an appropriate calculus-based justification for the fact that 
\[g\] is increasing?
Choose 1 answer:
Choose 1 answer:
(Choice A)   As the 
\[x\]-values increase, the values of 
\[g\] also increase.
A
As the 
\[x\]-values increase, the values of 
\[g\] also increase.
(Choice B)   
\[g'\] is decreasing.
B
\[g'\] is decreasing.
(Choice C)   
\[g'\] is positive.
C
\[g'\] is positive.

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.

Explore the calculus-based justification for why a function is increasing using derivatives. Learn how to interpret graphs of functions and their derivatives to determine increasing behavior. Suitable for advanced high school students.

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