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. $$ \begin{array}{l}{f(0)=f^{\prime}(0)=f^{\prime}(2)=f^{\prime}(4)=f^{\prime}(6)=0} \\ {f^{\prime}(x) > 0 \text { if } 0 < x < 2 \text { or } 4 < x < 6} \\ {f^{\prime}(x) < 0 \text { if } 2 < x < 4 \text { or } x > 6} \\ {f^{\prime \prime}(x) > 0 \text { if } 0 < x < 1 \text { or } 3 < x < 5} \\ {f^{\prime \prime}(x) < 0 \text { if } 1 < x < 3 \text { or } x > 5, \quad f(-x)=f(x)}\end{array} $$

$20-27$ Sketch the graph of a function that satisfies all of the given
conditions.
$$
\begin{array}{l}{f(0)=f^{\prime}(0)=f^{\prime}(2)=f^{\prime}(4)=f^{\prime}(6)=0} \\ {f^{\prime}(x) > 0 \text { if } 0 < x < 2 \text { or } 4 < x < 6} \\ {f^{\prime}(x) < 0 \text { if } 2 < x < 4 \text { or } x > 6} \\ {f^{\prime \prime}(x) > 0 \text { if } 0 < x < 1 \text { or } 3 < x < 5} \\ {f^{\prime \prime}(x) < 0 \text { if } 1 < x < 3 \text { or } x > 5, \quad f(-x)=f(x)}\end{array}
$$

Key Concepts

Graph Sketching Techniques

Sketching the graph of a function involves combining information about symmetry, critical points, monotonicity, and concavity. By analyzing where the function increases or decreases and where it bends upwards or downwards, one can accurately depict the overall shape and key features of the graph.

Concavity and Inflection Points

Concavity describes whether a function curves upward or downward and is dictated by the sign of the second derivative. A positive second derivative indicates that the function is concave up, while a negative second derivative shows concave down behavior. Points where the concavity changes are known as inflection points and are critical for understanding the curvature of the graph.

Monotonicity

Monotonicity refers to the behavior of a function in terms of increasing or decreasing trends over certain intervals. The sign of the first derivative indicates whether the function is rising (positive derivative) or falling (negative derivative), which is fundamental for mapping the function's progression.

Critical Points

Critical points occur where a function's derivative is zero or undefined. These points are essential in determining where local maxima, minima, or saddle points may exist, and they provide key landmarks in understanding the overall graph.

Even Functions

An even function satisfies the property f(-x) = f(x) for all x in its domain. This type of symmetry about the y-axis simplifies the process of sketching a graph since the behavior on one side of the y-axis mirrors the other.

Transcript

00:01 So in this question, i'm given a lot of information and asked to draw a graph of a function.

00:07 I'm told f of zero is equal to f prime of zero, equal to f prime of two, f prime of four, f prime of six, and they're all equal to zero.

00:16 And i've turned the information and the question into a sign test of the first derivative and the second derivative.

00:23 So the first thing i see, oh, i also have a little piece of information that the function is odd, where they tell me f of negative x is equal to.

00:32 I believe it is odd.

00:34 No, it's even.

00:35 F of negative x is equal to f of x.

00:38 So this tells me that's the even function rule that i am dealing with an even function.

00:44 Okay, so f of zero is zero.

00:46 That's great.

00:47 That means the function goes to the origin.

00:49 And now i'll mark one, two, three, four, five, six.

00:53 In the other side, i'll go down to negative six, three, four, negative six.

00:57 And i'm going to draw the right and just copy it to the left, mirror it over the y -axis because the function's even.

01:02 So basically i have a relative maximum followed by a minimum followed by a maximum.

01:12 And then i have a point of inflection at 1, 3, and 5.

01:18 So i have to increase between 1 and 2 where i hit a relative maximum.

01:25 Then i'm going to come down and hit a relative minimum at 4.

01:29 I might as well just put it on the x -axis...

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