Use a sign chart for f^' to determine the intervals on which...

Use a sign chart for $f^{\prime}$ to determine the intervals on which each function $f$ is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.
$$
f(x)=\frac{e^{x}}{1+e^{x}}
$$

Key Concepts

Sign Chart Analysis

A sign chart is a visual tool used to analyze the sign (positive or negative) of the derivative over different intervals. By dividing the domain of the function based on critical points, one can test the sign of the derivative in each interval. This analysis helps determine whether the function is increasing (derivative > 0) or decreasing (derivative < 0) in each interval.

Monotonicity

Monotonicity refers to the property of a function being entirely non-decreasing or non-increasing over an interval. By determining the sign of the derivative, one can conclude which intervals a function is monotonically increasing or decreasing. This concept is central to understanding and graphing the behavior of functions in calculus.

Critical Points

Critical points are locations in the domain of the function where the derivative is zero or undefined. These points are important because they can indicate local maxima, minima, or inflection points. Identifying critical points is a key step in understanding the behavior of a function, especially in determining where the function increases or decreases.

Derivative

The derivative represents the instantaneous rate of change of a function. It is used to determine how a function is changing at any given point, and it serves as a fundamental tool in calculus for studying and analyzing functions. Computation of the derivative involves applying differentiation rules such as the product, quotient, or chain rules, depending on the structure of the function.

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