Analyzing Critical Numbers: Decoding the Extrema and Inflection Points of Mathematical Functions

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

FALSE…critical numbers are places where the derivative is zero or undefined. It is possible that it could be a local extrema or an inflection point, but it doesn’t necessarily have to be. Keep in mind that the derivative will be undefined at any discontinuity. Therefore c could have also been a point of removable discontinuity, jump discontinuity or infinite discontinuity.

If c is a critical number of f, then there are two possibilities:

1. f has a relative extrema at c: Either relative maximum or minimum.
When a function has a critical point at c, it means that the derivative of the function is either zero or undefined at c. In other words, the slope of the tangent line to the graph of f is either zero or does not exist at c. When the slope changes from positive to negative at c, f has a relative maximum at c. Conversely, when the slope changes from negative to positive at c, f has a relative minimum at c.

2. f has an inflection point at c:
If the second derivative of f changes sign at a critical point c, then f has an inflection point at c. This means that the concavity of the graph of f changes at c from being concave up to being concave down or vice versa.

To determine whether f has a relative maximum, relative minimum, or an inflection point at c, we need to analyze higher-order derivatives of f. If the second derivative of f at c is positive, then f has a relative minimum at c. Conversely, if the second derivative of f at c is negative, then f has a relative maximum at c. If the second derivative is zero, then further analysis is needed to determine the nature of the critical point at c.

In summary, if c is a critical number of f, then f has a relative maximum, relative minimum, or an inflection point at c, depending on how the first and second derivatives of f behave at c.

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