For each of the exercises listed below, suppose that the...

For each of the exercises listed below, suppose that the function that is graphed is not $f(x),$ but $f^{\prime}(x) .$ Find the open intervals where $f(x)$ is (a) increasing or (b) decreasing.

Exercise 7

Key Concepts

First Derivative Test

The first derivative test is a method used to determine where a function is increasing or decreasing by examining the sign changes of its derivative. This test identifies intervals where the function’s derivative is positive or negative and, consequently, whether the function itself is increasing or decreasing over those intervals.

Graphical Analysis of Derivatives

This concept involves analyzing the graph of the derivative of a function to deduce the behavior of the original function. By assessing the regions where the graph of the derivative lies above or below the x-axis, one can determine the intervals on which the function increases or decreases, respectively.

Relationship between a Function and its Derivative

This concept explains that the derivative of a function indicates the rate at which the function is changing. When analyzing whether a function is increasing or decreasing, the sign of its derivative is key: if the derivative is positive over an interval, the function is increasing on that interval, and if it is negative, the function is decreasing.

Monotonicity

Monotonicity refers to the property of a function being entirely non-increasing or non-decreasing over a specific interval. Determining the monotonicity of a function involves analyzing the sign of the derivative. A positive derivative suggests monotonic increase, while a negative derivative indicates a monotonic decrease.

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