The key here is the description of the behavior of the rate of change of the function $g$. The rate of change of a function is its first derivative, $g'(x)$. If the rate of change is increasing for $x < 2$ and decreasing for $x > 2$, it suggests that the second derivative, $g''(x)$, changes signs at $x = 2$.

To break it down:

• For $x < 2$: The rate of change is increasing, meaning $g'(x)$ is increasing, which implies $g''(x) > 0$. This indicates the graph is concave up.
• For $x > 2$: The rate of change is decreasing, meaning $g'(x)$ is decreasing, which implies $g''(x) < 0$. This indicates the graph is concave down.
• At $x = 2$: Since the concavity changes from positive (concave up) to negative (concave down), there is a point of inflection at $x = 2$.

Now, analyzing the given options:

1. The graph of $g$ has a minimum at $x = 2$:

   • This is not necessarily true because the behavior of the first derivative does not indicate a minimum at $x = 2$, only a change in concavity.

2. The graph of $g$ has a maximum at $x = 2$:

   • This is also not necessarily true for the same reason as above. We don’t have information about the first derivative behavior suggesting a maximum.

3. The graph of $g$ has a point of inflection at $x = 2$, is concave down for $x < 2$, and is concave up for $x > 2$:

   • This is incorrect because for $x < 2$, the rate of change is increasing, implying concave up behavior, not concave down.

4. The graph of $g$ has a point of inflection at $x = 2$, is concave up for $x < 2$, and is concave down for $x > 2$:

   • This is correct. The behavior of the second derivative changes from positive to negative, indicating a point of inflection at $x = 2$, with the concavity being up for $x < 2$ and down for $x > 2$.

Conclusion:

The correct answer is Option 4.

Would you like more details on any specific part of this explanation?

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Related questions:

1. How can we use the second derivative test to confirm a maximum or minimum?
2. What is the significance of a point of inflection in a function's graph?
3. How does concavity affect the shape of a graph?
4. What is the relationship between the second derivative and the concavity of a function?
5. How can you determine intervals of concavity for a polynomial function?

Tip: Always remember, a point of inflection occurs where the concavity of the graph changes, which is where the second derivative changes sign.