Use the given derivative to find all critical points of f...

Use the given derivative to find all critical points of $f$ and at each critical point determine whether a relative maximum, relative minimum, or neither occurs. Assume in each case that $f$ is continuous everywhere.
$$
f^{\prime}(x)=x^{2}\left(x^{3}-5\right)
$$

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Key Concepts

Relative Extrema

Relative extrema refer to the local maximum and minimum points of a function. These are the points where the function attains higher or lower values relative to surrounding points. Identifying relative extrema is crucial for understanding the optimized behavior of functions in various applications.

Sign Analysis

Sign analysis involves examining the signs (positive or negative) of a function’s derivative across different intervals to determine whether the function is increasing or decreasing in those intervals. This technique is essential for applying the first derivative test and ultimately for identifying the nature of critical points.

First Derivative Test

The first derivative test is a method used to classify critical points by analyzing the sign of the derivative before and after the point. A sign change from negative to positive indicates a relative minimum, while a change from positive to negative indicates a relative maximum. No sign change often means the critical point is neither a maximum nor a minimum.

Critical Points

Critical points are values in the domain of a function where its derivative is zero or does not exist. They are important because they indicate possible locations for local maxima, minima, or points of inflection, which are key to understanding the function’s behavior.

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