Understanding Critical Numbers: Examining the Relationship between Concavity and Extremes

If c is a critical number of f, then f has a relative maximum, relative minimum, or an inflection point at c

To determine whether a critical number of a function f corresponds to a relative maximum, relative minimum, or an inflection point, we need to consider the second derivative of the function

To determine whether a critical number of a function f corresponds to a relative maximum, relative minimum, or an inflection point, we need to consider the second derivative of the function.

First, let’s define what a critical number is. A critical number of a function f is a value c in the domain of f where either f'(c) = 0 or f'(c) is undefined.

Now, if c is a critical number of f, we need to evaluate the second derivative of f at c, denoted as f”(c).

If f”(c) > 0, then the function f is concave up at c. This means that the graph of f is shaped like a U around c, indicating a relative minimum at c.

If f”(c) < 0, then the function f is concave down at c. This means that the graph of f is shaped like an inverted U around c, indicating a relative maximum at c. If f''(c) = 0 or f''(c) is undefined, we cannot make any conclusions about the behavior of f at c. In this case, c may correspond to an inflection point of the function where the concavity changes. To summarize: - If f''(c) > 0, f has a relative minimum at c.
– If f”(c) < 0, f has a relative maximum at c. - If f''(c) = 0 or f''(c) is undefined, we cannot determine the nature of f at c without additional information. It's important to note that this is a general guideline, but there may be cases where additional analysis is required to determine the nature of the function at a critical point.

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