For the polynomial function g, the rate of change of g is increasing for x<2 and decreasing for x>2. which of the following must be true?

1, the graph of g has a minimum at x=2 
2. the graph of g has a maximum of x=2 
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
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

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.

For the polynomial function g, the rate of change of g is increasing for x<2 and decreasing for x>2. Which of the following must be true? 1. The graph of g has a minimum at x=2. 2. The graph of g has a maximum at x=2. 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. 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.

Understand how to determine the point of inflection and concavity changes in a polynomial function where the rate of change increases for x<2 and decreases for x>2. This example breaks down derivatives and inflection points, suitable for students in Grades 11-12 studying Calculus.

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