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.
$$
f^{\prime}(x)=x^{4}\left(e^{x}-3\right)
$$

Key Concepts

Derivative

The derivative of a function represents the instantaneous rate of change or the slope of the tangent line at any point on the function. It is a fundamental concept in calculus that provides insight into the behavior of a function, including where the function is increasing, decreasing, or staying constant.

Critical Points

Critical points are the values of the independent variable where the derivative is zero or undefined. They are important because they are potential locations of relative extrema (maximums or minimums) or inflection points. In the context of optimization, identifying these points is the first step in determining the local behavior of the function.

First Derivative Test

The First Derivative Test is a method used to classify critical points by analyzing the sign changes of the derivative around these points. By examining whether the derivative changes from positive to negative or vice versa, one can determine if a critical point is a relative maximum, minimum, or neither. This test is crucial for understanding the local behavior of functions where the derivative vanishes.

Sign Analysis of the Derivative

Sign analysis involves determining the sign (positive or negative) of the derivative in intervals defined by the critical points. This process helps to reveal where the function is increasing or decreasing. By coordinating this information with the First Derivative Test, one can effectively classify the nature of each critical point.

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