20-27 Sketch the graph of a function that satisfies all of...

$20-27$ Sketch the graph of a function that satisfies all of the given conditions. $$ \begin{array}{l}{f^{\prime}(0)=f^{\prime}(2)=f^{\prime}(4)=0} \\ {f^{\prime}(x) > 0 \text { if } x < 0 \text { or } 2 < x<4} \\ {f^{\prime}(x) < 0 \text { if } 0 < x<2 \text { or } x > 4} \\ {f^{\prime \prime}(x) > 0 \text { if } 1 < x<3, \quad f^{\prime \prime}(x) < 0 \text { if } x < 1 \text { or } x > 3}\end{array} $$

$20-27$ Sketch the graph of a function that satisfies all of the given
conditions.
$$
\begin{array}{l}{f^{\prime}(0)=f^{\prime}(2)=f^{\prime}(4)=0} \\ {f^{\prime}(x) > 0 \text { if } x < 0 \text { or } 2 < x<4} \\ {f^{\prime}(x) < 0 \text { if } 0 < x<2 \text { or } x > 4} \\ {f^{\prime \prime}(x) > 0 \text { if } 1 < x<3, \quad f^{\prime \prime}(x) < 0 \text { if } x < 1 \text { or } x > 3}\end{array}
$$

Key Concepts

Critical Points

Critical points occur where the first derivative of a function is zero or undefined. These points are important because they are potential locations for local maxima, local minima, or saddle points and therefore help in understanding the overall shape of the graph.

Increasing and Decreasing Intervals

The sign of the first derivative indicates whether a function is increasing or decreasing. When the first derivative is positive over an interval, the function is increasing there, and when it is negative, the function is decreasing. This concept helps in sketching the overall trend of the function without knowing its exact formula.

Concavity and Convexity

The sign of the second derivative determines the concavity of the function. A positive second derivative implies that the function is concave up (or convex), meaning the graph bends upward and the slope is increasing. Conversely, a negative second derivative indicates that the function is concave down, where the graph bends downward. Understanding concavity is pivotal in visualizing how the function curves between critical points.

Inflection Points

Inflection points are where the concavity of the function changes. This occurs when the second derivative changes sign. Identifying inflection points is crucial for accurately sketching the graph, as they mark transitions in the curvature of the function.

Graphical Sketching Using Derivative Information

Using the information provided by the first and second derivatives, one can piece together a rough sketch of a function. This involves marking critical points, determining intervals of increasing and decreasing behavior, and identifying regions of concavity and points of inflection. This comprehensive approach allows for an accurate representation of the function’s behavior even without an explicit formula.

Transcript

00:01 Hello everyone, so today we're going to sketch a graph based on this information at the top of our page.

00:06 So there's a lot to take in, so what i like to do at first is just draw a little number line.

00:10 Don't even worry about y yet.

00:12 Just try to take note of what's happening between each x interval, and then once we have that, it'll be easier to draw a graph.

00:18 So let's see.

00:20 First things first, we know that at x value 0, 2, and 4 is something special is happening.

00:25 The first derivative is 0, which means that our slope is going to be 0 at those points.

00:29 So first i'm just going to draw a little x equals 0, x equals 2, x equals 4 here.

00:37 And i'm going to draw these little, this is just my personal notation for now, but i'm just drawing a little dots with lines through them so that i know that the slope is zero at those points.

00:47 And we can keep looking at the rest of our information here.

00:50 So f prime of x is greater than 0 when x is less than 0 or when x is between 2 and 4.

00:58 So if f prime of x is greater than zero, we know that it is increasing.

01:03 So we know it should be increasing for x less than zero.

01:09 And between two and four, it is also increasing.

01:12 Let's make this like, let's make it like something fun.

01:17 Let's make it green.

01:19 Okay, sorry about that.

01:21 Okay.

01:22 So, right, so there's this bullet done.

01:26 We know f prime of x is less than zero.

01:31 Between x equals 0 and 2 and x greater than 4 so we know that if f prime of x is less than 0 it is decreasing so it should be decreasing between 0 and 2 and decreasing from 4 onward and okay that's it for the first derivative so we know where it's increasing and decreasing and where the slope is 0 now let's see about the second derivative the second derivative is greater than 0 between 1 and 3 okay, let's make some little extra x values there.

02:09 So we know between 1 and 3, if f double prime is greater than 0, that means our function is concave up.

02:18 And likewise, when, let's just do blue, if our function is where the second derivative of our function is less than 0, when x is less than 1, or greater, greater than three.

02:34 We know that ff double prime is negative, then our function is concave down at those points.

02:42 So we have a little compendium here of what our function looks like.

02:46 Let's take our notes that we have and try to make something of them...

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