In Exercises 17-42,( a ) find the critical numbers of f( if any ),(b) find the open interval(s)

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In Exercises 17-42,( a ) find the critical numbers of f( if...

In Exercises $17-42,(\text { a })$ find the critical numbers of $f(\text { if any }),(b)$ find the open interval(s) on which the function is increasing or decreasing, (c) apply the First Derivative Test to identify all relative extrema, and (d) use a graphing utility to confirm your results. $$ f(x)=\left\{\begin{array}{ll}{2 x+1,} & {x \leq-1} \\ {x^{2}-2,} & {x>-1}\end{array}\right. $$

In Exercises $17-42,(\text { a })$ find the critical numbers of $f(\text { if any }),(b)$ find the open interval(s) on which the function is increasing or decreasing, (c) apply the First Derivative Test to identify all relative extrema, and (d) use a graphing utility to confirm your results.
$$
f(x)=\left\{\begin{array}{ll}{2 x+1,} & {x \leq-1} \\ {x^{2}-2,} & {x>-1}\end{array}\right.
$$

Key Concepts

Graphical Analysis

Graphical analysis involves using a graphing utility or sketching the function to visually confirm analytical findings. It is an essential tool for verifying the changes in the function's behavior, such as the locations of critical points and the intervals of increase or decrease, which provides a comprehensive understanding of the function.

First Derivative Test

The First Derivative Test employs the information from the derivative's sign changes to classify critical points as local minima, local maxima, or neither. By examining how the derivative transitions around a critical number, one can deduce whether the function has reached a peak or a trough at that point.

Piecewise Functions

A piecewise function is defined by multiple sub-functions, each applying to a certain interval of the domain. Understanding how these different expressions work over different intervals is crucial for analyzing properties like continuity and differentiability at the transition points between the pieces.

Critical Numbers

Critical numbers are values in the function's domain where the derivative is zero or undefined. They are key in determining the local behavior of the function, as these points may correspond to local maxima, local minima, or inflection points, and are identified by calculating the derivative on each segment of the piecewise function.

Increasing and Decreasing Intervals

Determining the intervals on which a function is increasing or decreasing involves assessing the sign of the first derivative in various regions of the domain. If the derivative is positive on an interval, the function is increasing there; if negative, the function is decreasing. This concept is critical for mapping out the function's overall behavior.

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