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.
$$
f(x)=5-|x-5|
$$

Key Concepts

Absolute Value Functions

Absolute value functions involve the magnitude of a number regardless of its sign and are usually defined piecewise. This concept is crucial as such functions can exhibit sharp turns or cusps where the expression inside the absolute value changes sign, affecting differentiability and analysis of the function’s behavior.

Critical Numbers

Critical numbers are points in the domain of a function where the derivative is zero or does not exist. Identifying these points is essential for determining local behavior such as potential minima, maxima, or points of inflection, and they serve as key markers in the analysis of the function’s increasing and decreasing intervals.

Derivatives and Differentiability

The derivative measures the rate of change of a function and is used to assess local behavior. In the context of an absolute value or piecewise function, it is important to check for differentiability at points where the function’s formula changes, as these points might not have a derivative despite the function being continuous.

Increasing and Decreasing Intervals

Analyzing where a function’s derivative is positive or negative helps determine intervals on which the function is increasing or decreasing. These intervals are used to understand the function’s overall behavior and to locate relative extrema.

Relative Extrema

Relative or local extrema refer to the local maximum or minimum values of a function. They are found by evaluating the function at its critical numbers and using tests (such as the first derivative test) to confirm whether these points correspond to local highs or lows.

Graphical Analysis

Graphical analysis involves using plotting tools to visually verify the behavior of a function. This step is important for confirming calculated results, especially for functions with piecewise definitions or absolute value expressions, where visual cues help identify key features like cusps, turning points, and intervals of monotonicity.

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