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

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. $$ f(x)=5-|x-5| $$

Key Concepts

Critical Numbers

Critical numbers are values in the domain of a function where the derivative is either zero or does not exist. These points are important because they indicate where the function's behavior may change, such as shifting from increasing to decreasing or vice versa, and are essential for further analysis of extrema.

First Derivative Test

The First Derivative Test is used to determine the local extrema of a function by examining the sign changes of its derivative around the critical points. It involves checking whether the function transitions from increasing to decreasing (indicating a local maximum) or from decreasing to increasing (indicating a local minimum).

Increasing and Decreasing Intervals

Increasing and decreasing intervals are ranges of the domain where a function consistently grows or falls, respectively. By analyzing the sign of the derivative, one can determine these intervals which help in understanding the overall behavior and trends of the function.

Relative Extrema

Relative extrema refer to the local maximum or minimum values of a function. Identifying these points is crucial because they show where the function attains locally highest or lowest values, and they often provide insight into the function's overall structure and behavior.

Absolute Value Function Analysis

Absolute value functions often lead to piecewise definitions because their formulas change based on the input. This characteristic can create points where the derivative does not exist, which is particularly important when identifying critical numbers and analyzing continuity and differentiability.

Graphing Utility Verification

Using a graphing utility to verify analytical results is a practical strategy to confirm the behavior of a function. It provides a visual representation of the function's overall shape, including its extrema, intervals of increase or decrease, and key features identified through derivative analysis.

Transcript

00:01 So in this problem, what we're going to be doing is we're going to be taking a number of steps to use the first derivative test to determine any local maximum or minimum that exist within this function, f of x here.

00:17 So what the first derivative test does is by looking at the derivative of f of x, f prime of x, if we look at the open intervals, which we find from the critical numbers, by finding the zeros of f prime of x.

00:35 What we can do is we can look at on either side of the interval whether the sign of f prime of x changes when it goes across these critical numbers.

00:45 So if we see that we have a positive to negative change in f prime of x across a critical number across those intervals, then that tells us that we have a relative maximum.

01:01 And conversely, if we see, that f prime of x undergoes a change from negative to positive cross that interval.

01:13 Then we see that we have a relative minimum.

01:18 And if f of x doesn't change, that means we have neither a relative maximum or minimum.

01:25 So let's take a look at our problem here, and we can go through the steps that we need to take.

01:30 So first what we want to do is we want to find the critical numbers.

01:34 And the way that we're going to do that is we're going to take the derivative of f of x and then find where f prime of x is equal to zero.

01:43 So we have an absolute value here and just to refresh the derivative of an absolute value, a function with an absolute value, that will be as such.

01:55 So we have, we take the derivative of you, the absolute value of you with respect to x.

02:04 What we're going to have is u over the absolute value of u times the derivative of u with the respect to x.

02:20 So now if we're to go ahead and apply this to our function, we'll find that f prime of x is going to be equal to x minus five over the absolute value of x minus five.

02:42 And we can simplify this actually.

02:48 So we'll have a piecewise function where we have f prime of x equal to 1 at when x is less than 5...

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