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Let's break down this problem step by step.

We are given:
- A function \( g(x) = f(x+1) - f(x) \).
- A function \( h(x) = g(x+1) - g(x) \).
- The information that \( h(x) = -6 \) for all values of \( x \), implying that \( h(x) \) is constant and negative.

### Step 1: Interpret \( h(x) = g(x+1) - g(x) = -6 \)
The equation \( h(x) = -6 \) means that the difference between \( g(x+1) \) and \( g(x) \) is always -6. This tells us that \( g(x) \) is a linearly decreasing function because the change in \( g(x) \) between any two points is constant and negative. Specifically, \( g(x) \) has a constant slope of -6.

### Step 2: Analyze \( g(x) = f(x+1) - f(x) \)
Since \( g(x) \) represents the difference between the values of \( f \) at \( x+1 \) and \( x \), and \( g(x) \) is a decreasing function, this suggests that \( f(x) \) must be a concave down function. A concave down function has decreasing slopes, which means the difference between successive function values (i.e., \( f(x+1) - f(x) \)) will decrease as \( x \) increases.

### Step 3: Conclusion
Since \( h(x) \) is negative and constant, \( g(x) \) is decreasing, and because the rate of change of \( f(x) \) (represented by \( g(x) \)) is decreasing, \( f(x) \) must be concave down.

Thus, the correct answer is **D**:
- "Because \( h(x) \) is negative and constant, \( g(x) \) is decreasing, and the graph of \( f \) is concave down."

Would you like a further explanation of any of these steps?

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Here are 5 related questions for further practice:
1. What does it mean for a function to be concave down?
2. How does the behavior of \( g(x) \) relate to the concavity of \( f(x) \)?
3. How do you determine the concavity of a function given its second derivative?
4. If \( h(x) \) were positive and constant, how would that affect the behavior of \( g(x) \) and \( f(x) \)?
5. How can you use the first derivative to analyze increasing or decreasing behavior of a function?

### Tip:
Always think of \( g(x) = f(x+1) - f(x) \) as a difference that can provide insight into how fast \( f(x) \) is changing and whether it's increasing or decreasing.

The function f is not explicitly given. The function g is given by g(x) = f(x+1) - f(x). The function h is given by h(x) = g(x+1) - g(x). If h(x) = -6 for all values of x, which of the following statements must be true?

Explore how to analyze functions using differences between successive values. This problem involves understanding concavity and linear behavior with functions g(x) = f(x+1) - f(x) and h(x) = g(x+1) - g(x), where h(x) is constant at -6. Learn how g(x) and f(x) behave in such cases, suitable for Grades 10-12.

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