Problem

Sketch the graph of a function that satisfies all…

02:05

Problem 27 Medium Difficulty

Sketch the graph of a function that satisfies all of the given conditions

$ f'(0) = f'(2) = f'(4) = 0 $,
$ f'(x) > 0 $ if $ x < 0 $ or $ 2 < x < 4 $,
$ f'(x) < 0 $ if $ 0 < x < 2 $ or $ x > 4 $,
$ f"(x) > 0 $ if $ 1 < x < 3 $, $ f"(x) < 0 $ if $ x < 1 $ or $ x > 3 $

Answer

$f^{\prime}(0)=f^{\prime}(2)=f^{\prime}(4)=0 \Rightarrow$ horizontal tangents at $x=0,2,4$
$f^{\prime}(x)>0$ if $x<0$ or $2<x<4 \Rightarrow f$ is increasing on $(-\infty, 0)$ and (2,4)
$f^{\prime}(x)<0$ if $0<x<2$ or $x>4 \Rightarrow f$ is decreasing on (0,2) and $(4, \infty)$
$f^{\prime \prime}(x)>0$ if $1<x<3 \Rightarrow f$ is concave upward on (1,3)
$f^{\prime \prime}(x)<0$ if $x<1$ or $x>3 \Rightarrow f$ is concave downward on $(-\infty, 1)$
and $(3, \infty) .$ There are inflection points when $x=1$ and 3

Watch More Solved Questions in Chapter 4

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31

Problem 32

Problem 33

Problem 34

Problem 35

Problem 36

Problem 37

Problem 38

Problem 39

Problem 40

Problem 41

Problem 42

Problem 43

Problem 44

Problem 45

Problem 46

Problem 47

Problem 48

Problem 49

Problem 50

Problem 51

Problem 52

Problem 53

Problem 54

Problem 55

Problem 56

Problem 57

Problem 58

Problem 59

Problem 60

Problem 61

Problem 62

Problem 63

Problem 64

Problem 65

Problem 66

Problem 67

Problem 68

Problem 69

Problem 70

Problem 71

Problem 72

Problem 73

Problem 74

Problem 75

Problem 76

Problem 77

Problem 78

Problem 79

Problem 80

Problem 81

Problem 82

Problem 83

Problem 84

Problem 85

Problem 86

Problem 87

Problem 88

Problem 89

Problem 90

Problem 91

Problem 92

Problem 93

Video Transcript

Okay, so we're giving a bunch of conditions and we're gonna try and use these to sketch a pretty accurate graph of our function F. Of X. And so the first thing I'm gonna do is try and figure out if these points at 02 and four are actually minimums are maximum values. So if we look at values less than zero, you can see that's here are function or the derivative, our function is greater than zero. So we're increasing and then we are decreasing for values between zero and two. So we go from increasing and decreasing. So at zero we have a local maximum. So I'm just going to draw a point, we'll put it here and it's going to be a maximum. And then for this point at two we can look at points or values of X that are less than two but greater than zero. And that's this part. You can see we're decreasing and the values greater than two but less than four were increasing. So we're going from decreasing increasing. So this is going to be a minimum value. And then if we look at values less than four but greater than two were increasing. And if we look at values that are greater than four um we are decreasing and I actually forgot a parentheses on this part. So we're going from increasing to decreasing. So this is another maximum at X. Is equal to four. So I'm just gonna put a point over here and now what we wanna do is make sure that we are doing the right thing cavity. So between one and three we have our second derivative being greater than zero. So that means that We have we are concave up between one and 3. So that would be I guess I'll put a point put this in blue at one and at three and then between negative infinity and one, we are concave down and from three to infinity were also concave down. So what I'm gonna do is I'm gonna draw this graph. So we're concave down before zero. So we have to be concave down and then this is a maximum and then we're concave down until There or wherever we come again until one. And then we start to be concave up And we're going and then we're concave down at three and we are decreasing from four to infinity. So this would be a good sketch of our graph. Or concave down before this point at one we have a maximum at X is equal to zero, a minimum and X equals two and then a maximum X is equal to four. And then we're concave up between one and three and then concave down again from three to infinity. So this is a good sketch of our graph given the conditions

Kian M.

Oregon State University

Related Courses

Calculus 1 / AB

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 4

Applications of Differentiation

Section 3

How Derivatives Affect the Shape of a Graph

Problem