In Exercises 41-46, use a graphing utility to graph the...

In Exercises $41-46,$ use a graphing utility to graph the function. Locate the absolute extrema of the function on the given interval.
$$
f(x)=\left\{\begin{array}{ll}{2-x^{2},} & {1 \leq x<3} \\ {2-3 x,} & {3 \leq x \leq 5}\end{array} \quad[1,5]\right.
$$

Key Concepts

Critical Points and Endpoints Analysis

The process of finding absolute extrema typically involves identifying critical points, where the derivative is zero or undefined, as well as evaluating the function at the endpoints of the interval. This systematic approach ensures that all potential candidates for maximum and minimum values are considered, especially in functions where different rules might apply, such as in piecewise definitions.

Graphing Utilities

Graphing utilities are powerful tools that assist in visualizing the behavior of functions. They can confirm analytical findings by providing a visual representation of the function’s overall behavior, including locating peaks and valleys. These tools are particularly useful when working with piecewise functions or complex functions where manual calculations might be laborious or error-prone.

Piecewise Functions

A piecewise function is defined by different expressions in different parts of its domain. Understanding piecewise functions involves analyzing each segment separately where each rule applies over specified intervals, and then considering how they connect or differ at the boundary points. This concept is crucial when assessing the overall behavior of a function that doesn’t follow a single rule across its entire domain.

Absolute Extrema

Absolute extrema refer to the highest and lowest values that a function attains over a particular interval. The absolute maximum is the greatest value reached by the function, while the absolute minimum is the smallest. These are found by evaluating the function at critical points and the endpoints of the interval, ensuring a complete assessment over a closed domain according to the extreme value theorem.

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