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all values  at which H has a local minimum:
all local minimum values of H:

Looking at the graph, we can find the local minimums by observing where the curve dips downward before rising again. These are typically valleys in the graph.

### Identifying the local minimum:
- A local minimum occurs around **x = 2**. At this point, the curve dips to its lowest point before increasing again.

### Value of the local minimum:
- The local minimum value of the function **H(x)** at **x = 2** is approximately **-4**.

### Final Answer:
- **Local minimum occurs at**: \( x = 2 \)
- **Local minimum value**: \( H(2) = -4 \)

Would you like more details or clarifications on any part of this?

Here are five follow-up questions to explore:
1. What are the values where the function \( H(x) \) has local maxima?
2. Can you find the intervals where \( H(x) \) is increasing or decreasing?
3. How would you determine the inflection points of \( H(x) \)?
4. What is the end behavior of \( H(x) \) as \( x \to \infty \) or \( x \to -\infty \)?
5. What could be the derivative \( H'(x) \) based on the shape of the graph?

**Tip:** The first derivative test is useful for identifying local minima and maxima by observing changes in the sign of the slope.

all values at which H has a local minimum: all local minimum values of H:

Discover how to find the local minimum values of the function H(x) by analyzing the graph and applying calculus concepts. This includes using the first derivative test and identifying critical points where the function has local minima, such as at x = 2 with H(2) = -4.

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