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Which Statement Correctly Identifies a Local Minimum of the Graphed Function?

In the study of calculus, understanding the behavior of functions is crucial. One important concept is that of local minimums, which represent the lowest points on a graph within a specific interval. Identifying these local minimums correctly is essential for various applications, such as optimizing functions in real-world scenarios. In this article, we will discuss the correct statement that identifies a local minimum of a graphed function and provide a comprehensive understanding of this concept.

A local minimum occurs when a function reaches its lowest point within a specific interval. It is important to note that this point may not be the absolute minimum of the entire function, but rather the lowest point within a particular range. To correctly identify a local minimum of a graphed function, the following statement is accurate:

“A point on the graph of a function is a local minimum if and only if there exists an open interval around the point where the function is greater than or equal to the value of the function at that point.”

This statement emphasizes two key components: the existence of an open interval and the function’s values within that interval. Let’s break down these components further:

1. Existence of an open interval: A local minimum occurs within a specific interval, which means there must be an open interval around the point in question. An open interval excludes the endpoints, allowing the function to have values both greater and smaller than the point in question.

2. Function values within the interval: To identify a local minimum, the function’s values within the open interval must be greater than or equal to the value of the function at the point. This condition ensures that the point is indeed the lowest within that interval.

Frequently Asked Questions (FAQs):

Q: What is the difference between a local minimum and an absolute minimum?

A: A local minimum represents the lowest point within a specific interval, while an absolute minimum is the lowest point of the entire function.

Q: Can a function have multiple local minimums?

A: Yes, a function can have multiple local minimums if there are multiple intervals where the function reaches its lowest point.

Q: How can I identify local minimums on a graph?

A: Look for points where the graph changes from decreasing to increasing, as these points often indicate local minimums. Additionally, examine the behavior of the function within specific intervals.

Q: Are local minimums always visible on a graph?

A: Not necessarily. Local minimums can be hidden if they occur in regions not plotted on the graph or if they coincide with other points.

Q: Can a function have a local minimum without having a local maximum?

A: Yes, a function can have a local minimum without a local maximum. The presence of a local maximum depends on the behavior of the function within the specific interval.

In conclusion, correctly identifying a local minimum of a graphed function requires understanding the existence of an open interval around the point and the function’s values within that interval. By considering these factors, one can accurately pinpoint the lowest points within specific intervals, allowing for effective analysis and optimization of functions in various contexts.

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