For a function g(x), g''(3)=-8 indicates that g(x) is ____________ at x=3.

Group of answer choices

concave up

concave down

increasing

decreasing

To determine the behavior of the function \( g(x) \) at \( x = 3 \) given \( g''(3) = -8 \), we need to understand the significance of the second derivative, \( g''(x) \).

### Analysis
1. **Second Derivative and Concavity**:
   - The second derivative \( g''(x) \) provides information about the concavity of the function.
   - If \( g''(x) > 0 \), the function is concave up at that point.
   - If \( g''(x) < 0 \), the function is concave down at that point.

Given that \( g''(3) = -8 \):
- Since \( g''(3) \) is negative, the function \( g(x) \) is concave down at \( x = 3 \).

### Conclusion
The correct answer is:

\[ \text{concave down} \]

Would you like more details on any step of the analysis? Here are five related questions you might find useful:

1. What is the significance of the second derivative test in determining local extrema?
2. How do you interpret the sign of the second derivative in terms of the shape of the graph?
3. Can a function be increasing or decreasing while being concave down?
4. What are the differences between concavity and monotonicity of a function?
5. How do you find points of inflection for a function?

**Tip**: When analyzing concavity, always check the sign of the second derivative to determine if the function is concave up or concave down at a specific point.

Explore how to determine the concavity of a function g(x) at x = 3 with g''(3) = -8. Learn the significance of the second derivative in calculus and its role in identifying concave down functions.

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