Find the critical numbers of f (if any). Find the open...

Find the critical numbers of $f$ (if any). Find the open intervals on which the function is increasing or decreasing and locate all relative extrema. Use a graphing utility to confirm your results.
$$
f(x)=\frac{x^{2}-3 x-4}{x-2}
$$

Key Concepts

Critical Numbers

Critical numbers are values in the domain of a function where its derivative is either zero or undefined. They are important because they indicate possible locations of local maxima or minima, and identifying them is a key step in the process of optimization and function analysis.

Intervals of Monotonicity

Intervals on which a function is increasing or decreasing are determined by the sign of its derivative. These intervals help define the overall behavior of the function and are essential for understanding how the function progresses between critical points.

Relative Extrema

Relative extrema are points where a function attains local maximum or minimum values. They typically occur at critical numbers where the sign of the derivative changes, and they are important for optimization problems and understanding the function's behavior in its domain.

Rational Functions and Discontinuities

Rational functions, which are defined as the ratio of two polynomials, often have points of discontinuity. Understanding these discontinuities, including removable ones, is crucial in correctly determining the domain of the function and analyzing its overall behavior.

Graphical Analysis

Graphing utilities provide a visual representation of a function’s behavior, including features such as critical points, increasing and decreasing intervals, and relative extrema. This tool is useful for confirming analytical results and gaining a deeper understanding of the function's properties.

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