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...

Key Concepts

Graphical Analysis

Graphical analysis involves using tools such as graphing calculators or computer software to visually represent the function. This approach provides confirmation of analytical results and helps in understanding the overall behavior of the function, including its critical points, intervals of increase or decrease, and relative extrema.

First Derivative Test

The First Derivative Test is a method used to classify critical points based on the behavior of the function's first derivative. By checking whether the derivative changes sign around a critical number, one can conclude whether it corresponds to a relative maximum, a relative minimum, or neither.

Critical Numbers

Critical numbers are the values in the domain of a function where its derivative is either zero or undefined. They are significant because they are potential candidates for local maximums, minimums, or points of inflection, and play a central role in analyzing the behavior of real functions.

Intervals of Increase and Decrease

Intervals of increase and decrease refer to the ranges in the domain where the function's values are respectively rising or falling. These intervals are determined by evaluating the sign of the first derivative: a positive first derivative indicates the function is increasing, while a negative value indicates it is decreasing.

Relative Extrema

Relative extrema are the local maximum and minimum points of a function, where the function takes on its highest or lowest value in a neighborhood of a given point. They are of interest in optimization and curve sketching, and can be identified by analyzing changes in the sign of the first derivative around critical numbers.

Transcript

00:01 So in this problem, what we're doing is we're applying the first derivative test in order to find the relative maximum or minimum of this function as of x.

00:11 So when we go ahead and we divide the interval of this function according to some critical numbers, when we go ahead and test the derivative along those intervals, we have three different scenarios that we can find.

00:28 So if we find that f prime of x, so the derivative of f x, is greater than 0, this means that f of x is increasing along that interval.

00:42 And i'll denote increasing with an arrow up.

00:45 Conversely, if we have, if we find that f prime of x is less than zero, this means that f of x, our original function, is decreasing along that interval.

00:58 And if f of x, f prime of x, excuse me, is equal to zero, this means that f of x is constant, so no change.

01:13 So now what the first derivative test does is in order to find whether there's a relative maximum or minimum, you compare the change of f prime of x, whether it goes from negative to positive or positive negative.

01:30 So if we have f prime of x and it changes from positive to negative across a test interval, this means that we have a relative max in the function f of x.

01:49 And on the other hand, if we go from negative to positive in f prime of x across a test interval, this means that we have a relative minimum.

02:04 And if there's no change, then it's neither a relative maximum or minimum.

02:09 So we're going to go ahead and apply the first derivative test to this function here.

02:14 And what we first have to do is we have to find the critical numbers.

02:18 And the way that we do that is we're first going to take the derivative of this function, f of x.

02:23 So when you expand this function, it becomes a bit simpler to take the derivative of.

02:30 And once you do that, you'll have a function.

02:34 That looks like this.

02:39 So you have x cubed plus 3x squared minus 4.

02:49 Now when you take f prime of x the derivative you'll get 3x squared plus 6x.

03:02 And to find the critical numbers is to find where this function f prime of x is equal to 0.

03:09 And when we go ahead and solve for the possible values of 0, we'll find that the critical numbers are 0 and negative 2...

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