a. Find the open intervals on which the function is...

$\begin{equation} \begin{array}{l}{\text { a. Find the open intervals on which the function is increasing and }} \\ {\text { decreasing. }} \\ {\text { b. Identify the function's local and absolute extreme values, if }} \\ {\text { any, saying where they occur. }}\end{array} \end{equation}$ $$\begin{equation} f(x)=x^{1 / 3}(x+8) \end{equation}$$

$\begin{equation}
\begin{array}{l}{\text { a. Find the open intervals on which the function is increasing and }} \\ {\text { decreasing. }} \\ {\text { b. Identify the function's local and absolute extreme values, if }} \\ {\text { any, saying where they occur. }}\end{array}
\end{equation}$
$$\begin{equation}
f(x)=x^{1 / 3}(x+8)
\end{equation}$$

Key Concepts

First Derivative Test

The first derivative test is a method used to classify the critical points of a function as local minima or maxima. It involves analyzing the change in sign of the derivative around the critical points and is a key step in determining the function's overall behavior.

Intervals of Increase and Decrease

Determining the intervals where a function is increasing or decreasing involves examining the sign of its derivative across different subdomains. If the derivative is positive over an interval, the function is increasing there; if it is negative, the function is decreasing.

Local and Absolute Extreme Values

Local extreme values refer to points where the function attains a minimum or maximum relative to nearby points, while absolute extreme values are the absolute highest or lowest values the function takes on its entire domain. Identifying these requires evaluating the function at its critical points and boundaries if applicable.

Derivative

The derivative is a fundamental tool in calculus that measures the instantaneous rate of change of a function. It is used to determine the slope of the function at any given point, which is crucial for analyzing the behavior of the function such as identifying where it increases or decreases.

Critical Points

Critical points are values in the domain of a function where the derivative is either zero or undefined. They are important because they indicate potential locations for local extreme values (minima and maxima) or changes in the function’s increasing-decreasing behavior.

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