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This problem asks for the behavior of \( g(x) = \int_0^x f(t) \, dt \), given the graph of \( f(x) \), for the interval \( 0 < x < 3 \).

### Key Steps:
1. **Relation Between \( f(x) \) and \( g(x) \):**
   - The value of \( g(x) \) is the definite integral of \( f(t) \) from \( 0 \) to \( x \).
   - \( g'(x) = f(x) \): The derivative of \( g(x) \) is equal to \( f(x) \).
   - \( g''(x) = f'(x) \): The second derivative of \( g(x) \) is equal to the slope of \( f(x) \).

2. **Analyze the Graph:**
   - For \( 0 < x < 3 \), \( f(x) \) is positive (above the x-axis). Hence, \( g(x) \) is **increasing**.
   - The graph of \( f(x) \) is decreasing (the slope of \( f(x) \) is negative). This means \( g'(x) = f(x) \) is decreasing, and \( g(x) \) is **concave down**.

### Conclusion:
For \( 0 < x < 3 \), \( g(x) \) is **increasing and concave down**.

### Correct Answer:
**D) Increasing and concave down.**

Would you like me to explain any part of this in more detail or explore related concepts?  

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### 5 Related Questions:
1. How does the Fundamental Theorem of Calculus connect \( f(x) \) and \( g(x) \)?
2. What does the concavity of a function tell us about its behavior?
3. How can we determine intervals of increase or decrease for a function using derivatives?
4. Why does the sign of \( f(x) \) determine whether \( g(x) \) is increasing or decreasing?
5. How does the second derivative test relate to concavity?

### Tip:
Always consider the behavior of \( f(x) \) when analyzing \( g(x) = \int f(t) \, dt \). Positive \( f(x) \) means \( g(x) \) increases, and the slope of \( f(x) \) determines concavity.

Below is the graph of a function f(x). Let g(x) = ∫₀ˣ f(t) dt. Then for 0 < x < 3, g(x) is:
A) decreasing and concave up
B) increasing and concave up
C) decreasing and concave down
D) increasing and concave down
E) There is not enough information to conclude any of the above.

Explore the behavior of g(x) = ∫₀ˣ f(t) dt using the graph of f(x). Understand key calculus concepts like increasing/decreasing functions and concavity. Includes a detailed solution and explanation of the Fundamental Theorem of Calculus.

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