Understanding Critical Numbers in Mathematics: How the Derivative of a Function Can Reveal Important Information

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

If c is a critical number of f, it means that the derivative of f at c is either zero or undefined

If c is a critical number of f, it means that the derivative of f at c is either zero or undefined.

Let’s analyze the cases:

1. If the derivative of f at c is zero (f'(c) = 0):
– This means that the slope of the tangent line to the graph of f at the point (c, f(c)) is zero.
– If f”(c) > 0, it indicates that the graph of f has a relative minimum at (c, f(c)).
– If f”(c) < 0, it indicates that the graph of f has a relative maximum at (c, f(c)). - If f''(c) = 0 or f''(c) is undefined, this test is inconclusive, and we need to consider higher-order derivatives or other methods to determine the nature of the point. 2. If the derivative of f at c is undefined: - This can occur when there is a vertical tangent line or a sharp corner at the point (c, f(c)). - In this case, we cannot determine whether f has a relative maximum, relative minimum, or an inflection point at (c, f(c)). We would need to examine the behavior of the function graphically or consider higher-order derivatives. In summary, if c is a critical number of f, the nature of the point (c, f(c)) could be a relative maximum, a relative minimum, or an inflection point, depending on the behavior of the function around that point. The concavity of the function at c, determined by the second derivative, helps us identify whether it is a relative maximum or a relative minimum, while undefined or vertical tangent lines make it difficult to determine the nature of the point solely based on the first derivative.

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