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. $$ f(x)=\left\{\begin{array}{ll}{-x^{3}+1,} & {x \leq 0} \\ {-x^{2}+2 x,} & {x>0}\end{array}\right. $$

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.
$$
f(x)=\left\{\begin{array}{ll}{-x^{3}+1,} & {x \leq 0} \\ {-x^{2}+2 x,} & {x>0}\end{array}\right.
$$

Key Concepts

Critical Numbers

Critical numbers are values in the domain of a function where the derivative is zero or undefined. They are important in the study of optimization as they provide potential locations for local maximums or minimums, which can be further examined using derivative tests.

First Derivative Test

The First Derivative Test involves analyzing the sign of the derivative on either side of a critical number. If the function's derivative changes from positive to negative at a point, that point is a local maximum; if it changes from negative to positive, it is a local minimum. This test provides a systematic way to classify extrema in differentiable functions.

Increasing and Decreasing Functions

Determining intervals where a function is increasing or decreasing is done by examining the sign of its derivative. Where the derivative is positive, the function is increasing, and where it is negative, the function is decreasing. This method is essential for understanding the overall behavior or trend of the function across its domain.

Piecewise Functions

Piecewise functions are defined by different expressions over different portions of their domains. Analyzing such functions requires examining each segment separately, ensuring proper handling of the boundaries between pieces, which may involve checking for continuity and differentiability at the points where the defining expressions change.

Transcript

00:01 So in this problem, we're going to use the first derivative test in order to find any local maximum or minimum in the function f of x.

00:10 So the way that we do that is we take the derivative of f of x, and when we apply that derivative function f prime of x equal to zero, we'll find some critical numbers.

00:21 So this will tell us the test intervals where we'll take the value of f prime of x and either get a positive or negative value.

00:28 So what the first derivative test does is it looks at the test intervals, looks at the sign of f prime of x going from one to the next, to determine if that point is a relative maximum or minimum.

00:43 So if we see that f prime of x changes from positive to negative, then we have a relative maximum.

00:48 And conversely, if we see that f prime of x goes from negative to positive, this means we have a relative minimum.

00:53 So to start, we're going to look at our function that we have f of x, and go ahead and take the derivative of that.

01:02 And since we have a piecewise function, we will take the derivative of each part on the respective intervals.

01:11 So first, where x is less than or equal to 0, what we have is negative x cubed to minus 1, and when we take the derivative of that, we'll have negative 3x squared.

01:26 And again on the same interval, if we go ahead and now take the derivative of the second part of this piecewise function, we'll go ahead and get negative 2x plus 2.

01:45 And this is on the interval of x greater than 0.

01:52 So now what we can do is we can go ahead and find the critical numbers.

01:58 So we can see here that if we set f of x equal to 0, looking at this first top part, of the function here.

02:09 We'll see that we have a critical number at x equals 0.

02:13 And then from this equation here, we also have one at x equals 1.

02:19 So these are two critical numbers.

02:21 And now we can set up our test intervals.

02:25 So we're first starting from negative infinity.

02:30 And we're going all the way to 0 for our first interval.

02:34 And then we'll go from 0 to 1.

02:41 And then from 1 to infinity.

02:47 And so we're looking at the sign of f prime of x first...