Chabot Mathematics
§3.2 Concavity
& Inflection
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Review §
3.1
 Any QUESTIONS About
• §3.1 → Relative Extrema
 Any QUESTIONS
About
HomeWork
• §3.1 →
HW-13
Chabot College Mathematics
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
§3.2 Learning Goals
 Introduce Concavity (a.k.a. Curvature)
 Use the sign of the second derivative to
find intervals of concavity
 Locate and examine
inflection points
 Apply the second
derivatives test for
relative extrema
Chabot College Mathematics
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
ConCavity Described
 Concavity quantifies the Slope-Value
Trend (Sign & Magnitude) of a fcn when
moving Left→Right on the fcn Graph
MTH15 • BLUE
m≈−4.4
m≈−4.4
m = df/dx
m≈0
MTH15 • RED
3
3
2
2
1
1
0
0
-1
-1
-2
-2
-3
-3
-4
-4
-5
1
2
3
4
-5
Position, x
Chabot College Mathematics
4
1
2
3
Position, x
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
4
Chabot College Mathematics
5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
MATLAB Code
% Bruce Mayer, PE
% MTH-15 •11Jul133
% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m
%
% The data
blue =[2.2 0 -1.4 -4.4]
red = [-4.4 -1.4 0 2.2]
%
% the 6x6 Plot
axes; set(gca,'FontSize',12);
subplot(1,2,1)
bar(blue, 'b'), grid, xlabel('\fontsize{14}Position, x'),
ylabel('\fontsize{14}m = df/dx'),...
title(['\fontsize{16}MTH15 • BLUE',]), axis([0 5 5,3])
subplot(1,2,2)
bar(red, 'r'), grid, xlabel('\fontsize{14}Position, x'),
axis([0 5 -5,3]),...
title(['\fontsize{16}MTH15 • RED',])
set(get(gco,'BaseLine'),'LineWidth',4,'LineStyle',':')
ConCavity Defined
 A differentiable function f on a < x < b is
said to be:
… concave DOWN (↓)
if df/dx is DEcreasing
on the interval
…concave up if
df/dx is INcreasing
on the interval.
Chabot College Mathematics
6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Example  Graphical Concavity
 Consider the function f given in the
graph and defined on the interval (−4,4).
 Approximate all
intervals on which
the function is
INcreasing,
DEcreasing,
concave up,
or concave down
Chabot College Mathematics
7
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Example  Graphical Concavity
 SOLUTION
 Because we have NO equation for the
function, we need to use our best
judgment:
• around where the
graph changes directions
(increasing/decreasing)
• where the derivative of
the graph changes directions
(concave up or down).
Chabot College Mathematics
8
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Example  Graphical Concavity
 To determine where the function is
INcreasing, we look for the graph to
“Rise to the Right (RR)”
Rising
Chabot College Mathematics
9
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Example  Graphical Concavity
 Similarly, the function is DEcreasing
where the graph “Falls to the Right
(FR)”:
Falling
Chabot College Mathematics
10
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Example  Graphical Concavity
 Conclude that f is increasing on the
interval (0,4) and decreasing on the
interval (−4,0)
 Now
Examine
Concavity.
Falling to Rt
Chabot College Mathematics
11
Rising to Rt
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Example  Graphical Concavity
 A function is concave UP wherever its
derivative is INcreasing. Visually, we
look for where the graph is
“curved upward”,
or “Bowl-Shaped”
Similarly, A function is concave DOWN
wherever its derivative is DEcreasing.
Visually, we look for where the graph is
“curved downward”,
or “Dome-Shaped”
Chabot College Mathematics
12
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Example  Graphical Concavity
 The graph is “curved UPward” for values
of x near zero, and might guess the
curvature to be positive between −1 & 1
Chabot College Mathematics
13
f is ConCave UP
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Example  Graphical Concavity
 The graph is “curved DOWNward” for
values of x on the outer edges of the
domain.
f is ConCave DOWN
Chabot College Mathematics
14
f is ConCave DOWN
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Example  Graphical Concavity
 Thus the function is concave UP approximately
on the interval (−1,1) and concave DOWN on
the intervals (−4, −1) & (1,4)
f is ConCave DOWN
f is ConCave DOWN
f is ConCave UP
Bruce Mayer, PE
Chabot College Mathematics
15
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Inflection Point Defined
 A function has an
inflection point 50
40
at x=a if f is
30
continuous
20
10
and the
0
CONCAVITY
-10
of f CHANGES -20
-30
at Pt-a
MTH15 • Inflection Point
y = f(x)
ConCave UP
Inflection
Point
ConCave DOWN
-40
-50
-2
-1
0
1
2
3
4
5
6
x
Chabot College Mathematics
16
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
7
8
9
Chabot College Mathematics
17
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
MATLAB Code
% Bruce Mayer, PE
% MTH-15 • 10Jul13
% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m
%
% The Limits
xmin = -2; xmax = 9;
ymin =-50; ymax = 50;
% The FUNCTION
x = linspace(xmin,xmax,1000); y =(x-4).^3/4 + (x+5).^2/7;
yOf4 = (4-4).^3/4 + (4+5).^2/7
%
% The ZERO Lines
zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];
%
% the 6x6 Plot
axes; set(gca,'FontSize',12);
whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green
plot(x,y, 'LineWidth', 5),axis([xmin xmax ymin ymax]),...
grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y =
f(x)'),...
title(['\fontsize{16}MTH15 • Inflection Point',])
hold on
plot(4, yOf4, 'd r', 'MarkerSize', 9,'MarkerFaceColor', 'r',
'LineWidth', 2)
plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2)
set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:10:ymax])
hold off
Example  Inflection Graphically
change from concave
down to up
change from concave
up to down
 The function shown above has TWO
inflection points.
Chabot College Mathematics
18
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
2nd Derivative Test
 Consider a function 𝑓 for Which
𝑑2 𝑓 𝑑𝑥 2 is Defined on some interval
containing a critical Point 𝑐 (Recall that
𝑑𝑓 𝑑𝑥 𝑥=𝑐 = 0) Then:
𝑑2𝑓
𝑑𝑥 2
• If
> 0, then 𝑓 is Concave UP at
𝑥 = 𝑐 so 𝑐 is a Relative MIN
𝑑2𝑓
𝑑𝑥 2
• If
< 0, then 𝑓 is Concave DOWN
at 𝑥 = 𝑐 so 𝑐 is a Relative MAX
Chabot College Mathematics
19
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Example  Apply 2nd Deriv Test
 Use the 2nd Derivative Test
2
x
f x  
to Find and classify all
x 1
critical points for the Function
 SOLUTION
df ( x  1)  2 x  x 2 1

 Find the
dx
( x  1) 2
2
critical points
x + 2x
0=
by solving:
2
(x +1)
2
df
0
=
x
+ 2x
 f ' x   0
dx
0 = x ( x + 2)
Chabot College Mathematics
20
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Example  Apply 2nd Deriv Test
 By Zero-Products:
0  xx  2  x  0 OR x  2
 Also need to check for values of x that
make the derivative undefined.
2
df x  2 x
• ReCall the

st
1 Derivative:
dx ( x  1) 2
• Thus df/dx is UNdefined for x = −1, But the
ORIGINAL function is ALSO Undefined at
the this value
– Thus there is NO Critical Point at x = −1
Chabot College Mathematics
21
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Example  Apply 2nd Deriv Test
 Thus the only critical points are at −2 & 0
 Now use the second derivative test to
determine whether each is a MAXimum
or MINimum (or if the test is
InConclusive):
d2y
d  x2  2x 
 
2
2 
dx  dx  x  1 

x  1 2 x  2   x  2 x  2 x  1 1

4
x  1
2
Chabot College Mathematics
22
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Example  Apply 2nd Deriv Test
 Before expanding the BiNomials, note
that the numerator and denominator can
be simplified by removing a common
factor of (x+1) from all terms:


d f  x  1 2 x  2  x 2  2 x  2x  1 1

2
dx
x  14
2
2


d 2 f x  1 x  12 x  2  2 x 2  2 x

2
dx
x  1x  13

d 2 f x  12 x  2  2 x 2  2 x

3
2
dx
x  1
Chabot College Mathematics
23


Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Example  Apply 2nd Deriv Test
 Now expand BiNomials:
d f 2x  2x  2x  2  2x  4x
2


3
2
3
dx
(
x

1
)
x  1
2
2
2
 Now Check Value of f’’’(0) & f’’’(−2)
2
d f
f ' '  2   2
dx
x  2
2

 2  0
3
 2  1
x 0
2

 2  0
3
0  1
2
d f
f ' ' 0   2
dx
Chabot College Mathematics
24
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Example  Apply 2nd Deriv Test
 The
Derivative is
NEGATIVE at x = −2
2nd
• Thus the orginal fcn is ConCave
DOWN at x = −2, and a
Relative MAX exists at this Pt
d2 f
2
dx
 2
x  2
d2 f
2
dx
 2
x 0
 Conversely, 2nd Derivative is POSITIVE
at x = 0
• Thus the orginal fcn is ConCave UP at x = 0
and a Relative MIN exists at this Pt
Chabot College Mathematics
25
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Example  Apply 2nd Deriv Test
 Confirm by Plot →
 Note the relative
MINimum at 0,
relative MAXimum
at −2, and a
vertical asymptote
where the function is undefined at x=−1
(although the vertical line is not part of
the graph of the function)
Chabot College Mathematics
26
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
ConCavity Sign Chart
 A form of the df/dx (Slope) Sign Chart
(Direction-Diagram) Analysis Can be
Applied to d2f/dx2 (ConCavity)
 Call the ConCavity Sign-Charts “DomeDiagrams” for INFLECTION Analysis
ConCavity
Form
d2f/dx2 Sign
++++++
Critical (Break)
Points
Chabot College Mathematics
27
−−−−−−
a
Inflection
−−−−−−
b
NO
Inflection
++++++
c
Inflection
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
x
Example  Dome-Diagram
 Find All Inflection
y  f  x   3x 5  5 x 4  1
Points for
• Notes on this (and all other) PolyNomial
Function exists for ALL x
 Use the ENGR25 Computer Algebra
System, MuPAD, to find
• Derivatives
• Critical Points
Chabot College Mathematics
28
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Example  Dome-Diagram
 The Derivatives
 The ConCavity
Values Between
Break Pts
• At x = −1
• At x = ½
 The Critical Points
• At x = ½
Chabot College Mathematics
29
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
MyPAD Code
Chabot College Mathematics
30
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Example  Dome-Diagram
 Draw Dome-Diagram
ConCavity
Form
d2f/dx2 Sign
Critical (Break)
Points
−−−−−−
−−−−−−
0
NO
Inflection
++++++
1
Inflection
x
 The ConCavity Does NOT change at 0,
but it DOES at 1
• Since Inflection requires Change, the
only Inflection-Pt occurs at x = 1
Chabot College Mathematics
31
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Example  Dome-Diagram
15
10
y = f(x) = 3x5 - 5x4 - 1
 The
Fcn
Plot
Showing
Inflection
Point at
(1,y(1))
= (1,−3)
MTH15 • Dome-Diagram
5
0
(1,−3)
-5
-10
-15
-1.5
-1
-0.5
0
0.5
1
1.5
x
Chabot College Mathematics
32
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
2
2.5
Chabot College Mathematics
33
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
MATLAB Code
% Bruce Mayer, PE
% MTH-15 • 11Jul13
% XYfcnGraph6x6BlueGreenBkGndTemplate1306.m
%
% The Limits
xmin = -1.5; xmax = 2.5;
ymin =-15; ymax = 15;
% The FUNCTION
x = linspace(xmin,xmax,1000); y =3*x.^5 - 5*x.^4 - 1;
%
% The ZERO Lines
zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];
%
% the 6x6 Plot
axes; set(gca,'FontSize',12);
whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green
plot(x,y, 'LineWidth', 5),axis([xmin xmax ymin ymax]),...
grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x) =
3x^5 - 5x^4 - 1'),...
title(['\fontsize{16}MTH15 • Dome-Diagram',])
hold on
plot(1,-3, 'd r', 'MarkerSize', 10,'MarkerFaceColor', 'r',
'LineWidth', 2)
plot(zxv,zyv, 'k', zxh,zyh, 'k', 'LineWidth', 2)
set(gca,'XTick',[xmin:0.5:xmax]); set(gca,'YTick',[ymin:5:ymax])
hold off
Example  Population Growth
 A population model finds that the
number of people, P, living in a city, in
kPeople, t years after the beginning of
2010 will be:
Pt   t  9t  10t  105
3
2
 Questions
• In what year will the population be
decreasing most rapidly?
• What will be the population at that time?
Chabot College Mathematics
34
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Example  Population Growth
 SOLUTION:
 “Decreasing most rapidly” is a phrase
that requires some examination.
“Decreasing” suggests a negative
derivative.
 “Decreasing most rapidly” means a
value for which the negative derivative
is as negative as possible. In other
words, where the derivative is a MIN
Chabot College Mathematics
35
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Example  Population Growth
 Need to find relative minima of functions
(derivative functions are no exception)
where the rate of change is equal to 0.
d d
 “Rate of change in

Pt   0

dt  dt
the population

dé 2
derivative, set
ë3t -18t +10ùû = 0
dt
equal to zero”
6t -18 = 0
TRANSLATES
mathematically to
t 3
Chabot College Mathematics
36
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Example  Population Growth
 The only time at which the second
derivative of P is equal to zero is the
beginning of 2013.
• Need to verify that the derivative is, in fact,
negative at that point:
dP
P ' t   3t 2  18t  10
dt
dP
P' 3  3(3) 2  18(3)  10
dt t 3
dP
P' 3  27  54  10  17
dt t 3
Chabot College Mathematics
37

Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Example  Population Growth
 Thus the function is
3
2
P(t)
=
t
9t
+10t +105
decreasing most
rapidly at the
inflection point at
the beginning
of 2013:
 The Model Predicts 2013 Population:
P3  (3)3  9(3) 2  10(3)  105  81 k  People
Chabot College Mathematics
38
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
WhiteBoard Work
 Problems From §3.2
• P45 → Sketch Graph using General
Description
• P66 → Spreading a Rumor
Chabot College Mathematics
39
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
All Done for Today
Rememgering
ConCavity:
cUP & frOWN
Chabot College Mathematics
40
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
41
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
ConCavity Sign Chart
ConCavity
Form
d2f/dx2 Sign
++++++
Critical (Break)
Points
Chabot College Mathematics
42
−−−−−−
a
Inflection
−−−−−−
b
NO
Inflection
++++++
c
Inflection
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
x
Max/Min Sign Chart
Slope
df/dx Sign
Critical (Break)
Points
Chabot College Mathematics
43
−−−−−−
++++++
a
Max
−−−−−−
b
NO
Max/Min
++++++
c
Min
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
x
Chabot College Mathematics
44
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Chabot College Mathematics
45
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Chabot College Mathematics
46
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Chabot College Mathematics
47
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Chabot College Mathematics
48
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Chabot College Mathematics
49
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx
Chabot College Mathematics
50
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-14_sec_3-2_Concavity_Inflection_.pptx