Sketch the graph of f by hand and use your sketch to find...

Sketch the graph of $f$ by hand and use your sketch to find the absolute and local maximum and minimum values of $f .$ (Use the graphs and transformations of Sections 1.2 and $1.3 . )$ $f(x)=\left\{\begin{array}{ll}{1-x} & {\text { if } 0 \leqslant x<2} \\ {2 x-4} & {\text { if } 2 \leq x \leq 3}\end{array}\right.$

Sketch the graph of $f$ by hand and use your sketch to find the absolute and local maximum and minimum values of $f .$ (Use the graphs and transformations of Sections 1.2 and $1.3 . )$
$f(x)=\left\{\begin{array}{ll}{1-x} & {\text { if } 0 \leqslant x<2} \\ {2 x-4} & {\text { if } 2 \leq x \leq 3}\end{array}\right.$

Key Concepts

Local Extreme Values

Local extreme values refer to points where a function reaches a maximum or minimum within a neighborhood, rather than on the entire domain. These points, found by setting derivatives to zero or checking for sign changes in the slope, provide insights into the function's behavior on restricted intervals.

Endpoint Analysis

In the context of functions defined on closed or partially closed intervals, analyzing endpoints is crucial because extreme values can occur at the boundaries even if derivatives do not indicate such. This examination ensures that all potential candidates for maxima or minima are considered, particularly in piecewise contexts where behavior changes at interval limits.

Absolute Extreme Values

Absolute maximum and minimum values are the highest and lowest values a function attains over its entire domain. Finding these values typically involves comparing the function values at critical points and endpoints, ensuring that no other point in the entire domain achieves a higher or lower output.

Piecewise Functions

A piecewise function is defined by different expressions over various intervals of the domain. It requires analyzing each segment separately because the behavior, such as slopes or discontinuities, can vary between the intervals. Understanding the overall function involves evaluating each piece under its specific conditions.

Graphing Piecewise Functions

Graphing these functions entails plotting each segment individually according to its specified domain. This process involves correctly marking endpoints (using open or closed circles as appropriate), applying any shifts, stretches, or reflections, and then combining the segments to illustrate the complete behavior of the function.

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