
Which statement is true about the local minimum of the graphed function?

Over the interval [–4, –2], the local minimum is 0.

Over the interval [–2, –1], the local minimum is 25.

Over the interval [–1, 4], the local minimum is 0.

Over the interval [4, 7], the local minimum is -7.

To determine which statement is true about the local minimum of the graphed function, we need to analyze each interval and the local minimum values given. We can evaluate each statement based on typical properties and descriptions of functions.

1. **Over the interval \([-4, -2]\), the local minimum is 0.**
   - This interval \([-4, -2]\) should be checked to see if there is a point where the function reaches a value of 0. For a typical function, it would make sense if it crosses or touches the x-axis.

2. **Over the interval \([-2, -1]\), the local minimum is 25.**
   - A minimum value of 25 in this interval suggests the function does not touch the x-axis within this interval but has a local minimum at a much higher point. We need to see if the function achieves a value of 25 at any point between \([-2, -1]\).

3. **Over the interval \([-1, 4]\), the local minimum is 0.**
   - If the function touches or crosses the x-axis within this interval, the local minimum might be at 0. This seems plausible as many functions could have a local minimum at the x-axis.

4. **Over the interval \([4, 7]\), the local minimum is -7.**
   - If the function reaches a negative value, particularly -7, in this interval, this could be the true minimum. The function would dip down to -7 at some point in the interval \([4, 7]\).

To identify the true statement, consider the possible function graph behavior:

### Analysis

1. **Over the interval \([-4, -2]\), the local minimum is 0.**
   - If the function does not cross the x-axis within this interval, then 0 is not the local minimum. However, if the function touches or crosses the x-axis at one point within \([-4, -2]\), this could be true.

2. **Over the interval \([-2, -1]\), the local minimum is 25.**
   - If the function is significantly high (at 25), it implies the local minimum is much above the x-axis within this small interval. This is unusual unless we have a specific graph like a steep parabola.

3. **Over the interval \([-1, 4]\), the local minimum is 0.**
   - This is likely true if the function crosses the x-axis at some point within \([-1, 4]\). Many graphs show such behavior within a wider range.

4. **Over the interval \([4, 7]\), the local minimum is -7.**
   - If the function dips down to -7 in this interval, this is the local minimum value. This can be true if the function drops significantly within this interval.

Considering common function behaviors and the intervals given, the statement that seems the most plausible based on typical function graph properties is:

**Over the interval \([-1, 4]\), the local minimum is 0.**

This is likely true for many functions that cross or touch the x-axis within a broad interval.

Would you like further details or have any other questions?

Here are some more questions you might want to ask:
1. How to find the local minimum of a function analytically?
2. How to interpret the behavior of a function from its graph?
3. What are the different types of local extremum in a function?
4. How to determine intervals of increase and decrease in a function?
5. What role do derivatives play in finding local minima and maxima?

**Tip:** When evaluating function properties such as local minima and maxima, always consider the behavior within the given intervals and check for any points where the function crosses or touches the x-axis.

Explore and understand local minima in function graphs through interval analysis. Learn how to determine the true statement about local minima over different intervals. Suitable for high school students studying functions and graph analysis.

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