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

step 1 So we have a price -wise function that is for minus x squared for interval x is less than or equ

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. $$ f(x)=\left\{\begin{array}{ll}{4-x^{2},} & {x \leq 0} \\ {-2 x,} & {x>0}\end{array}\right. $$

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.
$$
f(x)=\left\{\begin{array}{ll}{4-x^{2},} & {x \leq 0} \\ {-2 x,} & {x>0}\end{array}\right.
$$

Key Concepts

Relative Extrema

Relative extrema are the local maximum or minimum values that a function attains within a particular interval. Identifying these points is essential for understanding the overall shape and behavior of the function.

First Derivative Test

The First Derivative Test is a method used to determine local extrema by examining the behavior of the derivative around critical points. A change in the sign of the derivative around a critical point helps in classifying the point as a relative maximum or minimum.

Graphical Analysis

Graphical analysis involves using graphing utilities or sketches to visualize the function's behavior. This approach validates analytical findings by displaying features such as critical points, intervals of increase or decrease, and relative extrema.

Piecewise Functions

Piecewise functions are defined by different expressions on different intervals of the domain. When working with such functions, it is crucial to analyze each piece separately, especially at the boundaries where the definition changes, to ensure accurate interpretation of the function’s behavior.

Critical Numbers

Critical numbers are values in the domain of a function where its derivative is zero or does not exist. They are essential in identifying potential points of local maximums, minimums, or other significant changes in the behavior of the function.

Increasing/Decreasing Intervals

Increasing and decreasing intervals describe where a function's values rise or fall as the input increases. The determination of these intervals is commonly conducted by analyzing the sign of the function's first derivative, which indicates the rate of change.

Please give Ace some feedback