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

step 1 So we have an F of x function, it's a piecewise function. From interval less than equals to 1, w

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. $$ f(x)=\left\{\begin{array}{ll}{3 x+1,} & {x \leq 1} \\ {5-x^{2},} & {x>1}\end{array}\right. $$

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.
$$
f(x)=\left\{\begin{array}{ll}{3 x+1,} & {x \leq 1} \\ {5-x^{2},} & {x>1}\end{array}\right.
$$

Key Concepts

Critical Numbers

Critical numbers are values in the domain of a function where its derivative is either zero or undefined, provided the function is defined. These points are important because they are potential locations of relative extrema, where the function could change direction from increasing to decreasing or vice versa.

Increasing and Decreasing Intervals

Determining where a function is increasing or decreasing involves analyzing the sign of its first derivative over open intervals. If the derivative is positive over an interval, the function is increasing there; if negative, the function is decreasing. This concept is essential in understanding the overall behavior and trends of the function.

First Derivative Test

The First Derivative Test is used to classify critical numbers by assessing the sign change of the derivative around these points. If the derivative changes from positive to negative at a critical number, the function has a relative maximum; if it changes from negative to positive, the function has a relative minimum. This test is a key tool for identifying and confirming local extrema.

Piecewise Function Analysis

Piecewise functions are defined by different expressions over different intervals, which can lead to unique points of interest, especially at the boundaries where the formula changes. Analyzing such functions often requires careful consideration of continuity, differentiability, and matching conditions at the boundary points to ensure a complete understanding of the function’s behavior.

Graphing Utility in Calculus

Using a graphing utility is a practical way to confirm analytical findings about a function's behavior. It provides visual insights into the location of critical numbers, the shape of the graph over different intervals, and the presence of any extreme points or discontinuities. Graphing tools help in validating the results obtained from derivative tests and interval analysis.

Please give Ace some feedback