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Chabot Mathematics
§7.3 2Var
Optimization
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
[email protected]
Chabot College Mathematics
1
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Review §
7.2
Any QUESTIONS About
• §7.2 → Partial Derivatives
Any QUESTIONS
About
HomeWork
• §7.2 → HW-05
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§7.3 Learning Goals
Locate and classify relative extrema for
a function of two variables using the
second partials test
Examine applied problems
involving optimization of
functions of two variables
Discuss and apply the extreme value
property for functions of two variables to
find absolute extrema on a closed,
bounded region
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Local Maximum & Minimum
DEFINITION
A function of two variables has a local
maximum at (a,b) if f(x y) ≤ f(a,b) when
(x,y) is near (a,b). [This means that
f(x,y) ≤ f(a,b) for all points (x,y) in some
DISK with center (a,b).] The number
f(a,b) is called a local maximum value.
If f(x,y) ≥ f(a,b) when (x,y) is near (a,b),
then f(a,b) is a local minimum value.
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Local MaxMin Illustrated
The DISK centered
at (x,y)=(a,b)
produces a “Hill”
with peak at f(a,b)
Local (and Absolute)
max/min for z = f(x,y)
f a , b
z f x, y
a , b
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Quick Example
The function of x,y below has a
maximum of about 0.5 at approximately
(0.6, 0)
Relative Maximum
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CriticalPoints and Extrema
DEFINITION: A point (a,b) in the
domain of f(x,y), for which the first order
partial derivatives of f(x,y) exist, the
point (a,b) is a CRITICAL POINT if:
f x a , b
f
x
xa
y b
0
&
f
y
xa
0 f y a , b
y b
That is, BOTH Partials must equal Zero
at the Same Time
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CriticalPoints and Extrema
THEOREM: If f has a local maximum or
minimum at (a,b) then point (a,b) MUST
be a Critical Point at which both Partials
Simultaneously equal Zero.
While ALL max/min (Extrema) occur at
Critcal Points (CPs), NOT all CPs are
Extrema Points
• A Surface that contains a CP that is NOT
an Extremum is called a Saddle Surface
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Example Find Critical Points
4
2
Find All Critical Points for f ( x , y ) xy
x y
Critical points occur for a function of two
variables wherever both 1st Partials = 0
f
4
• For the given
y 2
2-Variable function x
x
&
f
y
x
Setting BOTH partials to Zero
Generates 2-Eqns in 2-Unknwns
f
x
0 y
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4
x
2
&
f
y
0x
2
y
2
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2
y
2
Example Find Critical Points
4
4
Using the x-Partial: 0 y x 2 y x 2
Now SubStitute into the y-Partial
0 x
2
y 2
Or x
2
4
2
x
4
2
0 x
4
x
BackSubbing:
4
Then
x 8
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2 2
3
y
x
3
2
1
83
0 x
4
2 41
x
x
16
x
1
3
x
2
1
1
x
2
x
2 2
3
8
x 2
y
4
2
2
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y 1
Example Find Critical Points
SOLUTION
Thus the only relative extremum of the
function occurs at (2,1)
MTH16 • Bruce Mayer, PE
Whether this
extremum is a
maximum,
minimum, or
neither is not yet
known. Its graph
suggests a minimum:
12
z = f(x,y)
11
10
9
8
7
6
0.5
1
1.5
1
2
2.5
3
3.5
0.5
MTH15 3Var 3D Plot.m
x
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y
1.5
Saddle Surface
At the “Saddle
Point” (0,0,0)
z
x
0
0
0
z
y
0
0
But, the Curve is a
• MINIMUM in the
xz plane
• MAXIMUM in the
yz plane
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Critical Point Condition
A Critical Point
ALWAYS
marks the
location of
a “Flat” Tangent
Plane, and can be
one of
• A MAXimum
• A MIMimum
• NEITHER
– i.e.; a SADDLE point
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The Nature of a CP
can (usually) Be
determined by the
Second Partials
Test
Assume for f(x,y) that
all needed Partial
exist then let
D x, y
f
2
x
2
f
2
y
2
f
x
y
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2
2
2nd Partials Test Procedure
Find a Critical Point (a,b) such that
f x a , b
f
x
xa
y b
0
&
f
y
xa
y b
0 f y a , b
Evaluate the “Discriminant” fcn, D(x,y)
from last slide, at the CP. That is, find
2
f
D a , b
x 2
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xa
y b
2
f
y 2
xa
y b
2
f
xy
xa
y b
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2
2nd Partials Test Procedure
If D(a,b) is NEGATIVE, then (a,b) is a
SADDLE POINT
2
f
For D(a,b)
f xx a , b
2
x x a
POSITIVE calc
y b
• If fxx(a,b) is POSITIVE, then (a,b) is a MAX
• If fxx(a,b) is NEGATIVE, then (a,b) is a MIN
If D(a,b) = 0 then the test is Inconclusive
• The pt (a,b) can be of ANY of the Three
forms; max, min, saddle
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Quick Example
For the Previous
Example Calc
f xx
4
y 2
x
x
f yy
2
x 2
y
y
f xy
2
x 2
x
y
Then D
f
D x, y
2
2
x
y
8
xy
3
8 4
2
x
D 1, 2 3 3 1
1 2
4
4
3
D 1, 2 8 1 4 1 3
y
8
8
And f xx 2 ,1 3 1
1
2
Now D>0 & fxx>0 so
f
2
f
2
(2,1) is a MAX
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2
2
Example Find Max Revenue
The Gladiator Goodies Company sells
Jumbo Cashews and the popular “Trial
by Trail” RaisinNut mix. Then GG
Industrial Engineering develops
Regression models for the products
q C 8 0 .5 x 0 .05 y ; qC in kCans
q R 6 0 . 2 x 0 . 2 y ; qR in kBags
• x ≡ Cashew Price in $ per Can
• y ≡ RaisinNut Price in $ per Bag
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Example Find Max Revenue
For this Financial Model
• Find a revenue function,
• determine at what prices revenue is
maximized, and
• find the maximum revenue from the sale of
these two products.
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Example Find Max Revenue
SOLUTION
The Company Revenue is the sum of
revenue from the Two products, so
R ( x, y ) pC qC p R q R
= x (8- 0.5x + 0.05y) + y ( 6 + 0.2x - 0.2y)
8 x 6 y 0 . 25 xy 0 . 5 x 0 . 2 y
2
2
To Maximize R, take the Partials and
set them to Zero
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Example Find Max Revenue
The Partial Derivatives
R x 8 0 . 25 y x
&
R y 6 0 . 25 x 0 . 4 y
0 8 0 . 25 y x
&
0 6 0 . 25 x 0 . 4 y
x 8 0 . 25 y
&
1 . 6 y 24 x
Combine the Two Equations
x 1 . 6 y 24 8 0 . 25 y x
1 . 6 y 0 . 25 y 8 24
y 32 / 1 . 35 23 . 70
BackSub to find: x 8 1 .25 ( 23 .70 ) 13 .93
So have ONE Critical Point at About
(13.93 $/can, 23.70 $/bag)
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Example Find Max Revenue
Next Find the Discriminant Function
2
2
f f
f
D x, y
2
2
x
y
xy
2
2
1 0 . 4 0 . 25 0 . 3375 0
2
The above calculation along with the
fact that ∂2R/∂x2 = −1 (<0) shows that
the critical point is a maximum
Then the Revenue at max
R max 13 . 93 , 23 . 7 8 13 . 93 6 23 . 7 0 . 25 13 . 93 23 . 7 0 . 5 13 . 93 0 . 2 23 . 7
2
R max 13 . 93 , 23 . 7 $ 126 815
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2
Constrained Domain Extrema
DOMAIN of many RealWorld 2Var Math
Models are Constrained For Various
Practical Reasons; Call This Domain, R
In a finite Constrained-Domain an
ABSOLUTE Max or Min is Present
• The Absolute Extrema exists at ONE of
– The EDGES, or BOUNDARY, of the Domain
Region, R
– The INTERIOR of R, at a Critical Point of the
2Var Function
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Constrained Domain Illustrated
Consider a 2Var MathModel, z = f(x,y)
with x & y constrained in the XY-Plane
by the 2D function g(x,y)=k (k a const)
as Illustrated below.
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Example Production Constraints
Gladiator Goodies Company (GGC)
Factory does NOT have Unlimited
Production Capacity.
The Factory has been Engineered to
this Design Constraint:
[No. RaisinNut Bags] ≥ 2·[ No. Cashew Cans]
• e.g., if the factory produces 2300 Cashew
Cans per week, then at least 4600
RaisinNut Bags also come off the Line
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Example Production Constraints
Recall the GGC Revenue MathModel as
developed by the Industrial Engineer
R ( x , y ) 8 x 6 y 0 . 25 xy 0 . 5 x 0 . 2 y
2
2
Given the Factory-Production
Constraint, Find the Maximum Revenue
that may be realized using the Cashew
and RaisinNut Production Line
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Example Production Constraints
The Goal is the as in the Previous
example; to Maximize Revenue. Stated
Mathematically →
q R 2 qC
(6 + 0.2x - 0.2y) ³ 2(8- 0.5x + 0.05y)
-10 +1.2x ³ 0.3y
y 4 x 100 3
A further constraint is that GGC will
NOT Give Away their products; thus
x0
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&
y 0
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Example Production Constraints
Then the THREE Constraints:
y 0
&
The
Constrained
Domain
Region
Graphically
27
y 4 x 100 3
45
40
35
30
25
20
15
10
• Practical
Price Region 5
0
in Dark Blue 0
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&
MTH16 • Bruce Mayer, PE
y = RaisinNut Price ($/Bag)
x0
MTH15 Quick Plot BlueGreenBkGnd 130911.m
5
10
15
20
x = Cashew Price ($/Can)
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MATLAB Code
% Bruce Mayer, PE
% MTH-15 • 01Aug13 • Rev 11Sep13
% MTH15_Quick_Plot_BlueGreenBkGnd_130911.m
%
clear; clc; clf; % clf clears figure window
%
% The Domain Limits
xmin = 0; xmax = 20;
% The FUNCTION **************************************
x = linspace(xmin,xmax,10000); y1 = 4*x-100/3; yFilter = (y1>0);
y=y1.*yFilter;
% ***************************************************
% the Plotting Range = 1.05*FcnRange
ymin = min(y); ymax = max(y); % the Range Limits
R = ymax - ymin; ymid = (ymax + ymin)/2;
ypmin = ymid - 1.025*R/2; ypmax = ymid + 1.025*R/2
%
% The ZERO Lines
zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ypmin*1.05 ypmax*1.05];
%
vxR = [xmax,xmax]; vyR = [0,ymax]; % close the constraint line at Right
% the 6x6 Plot
axes; set(gca,'FontSize',12);
whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green
area(x,y, 'LineWidth', 4),grid, axis([xmin 1.05*xmax ypmin ypmax]),...
xlabel('\fontsize{14}x = Cashew Price ($/Can)'), ylabel('\fontsize{14}y
= RaisinNut Price ($/Bag)'),...
title(['\fontsize{16}MTH16 • Bruce Mayer, PE',]),...
annotation('textbox',[.53 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor',
'none', 'String', 'MTH15 Quick Plot BlueGreenBkGnd 130911.m','FontSize',7)
hold on
plot(zxv,zyv, 'k', zxh,zyh, 'k', vxR,vyR, 'k', 'LineWidth', 2)
hold off
Example Production Constraints
Now consider all boundary points at
which critical values can occur. First,
consider the vertical line at x = 0:
R x , y 8 x 6 y 0 . 25 xy 0 . 5 x 0 . 2 y
2
R 0 , y 6 y 0 . 2 y
2
2
Taking dR(0,y)/dy = 0 produces a
Maximum at
6
y
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0 .4
15
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Example Production Constraints
Now consider all boundary points at
which critical values can occur. First,
consider the vertical line at x = 0:
R(x, y) = 8x + 6y + 0.25xy - 0.5x - 0.2y
2
2
R(0, y) = 6y - 0.2y2
Taking dR(0,y)/dy = 0 produces a
Maximum at
6
y
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0 .4
15
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Example Production Constraints
Next consider the horizontal line y = 0:
R x , y 8 x 6 y 0 . 25 xy 0 . 5 x 0 . 2 y
2
R x ,0 8 x 0 .5 x
2
2
8
Taking dR(x,0)/dx = 0
x
8
1
Results in an x-maximum
100
Lastly examine the Slanted Line: y 4 x
3
Sub the Slanted Line Constraint into the
Revenue Function
R ( x , y ) 8 x 6 y 0 . 25 xy 0 . 5 x 0 . 2 y
2
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2
Example Production Constraints
With
y 4x
100
3
100
R x ,4 x
R x , 4 x 33 . 33
3
100
100
100
2
8 x 6 4 x
0 . 25 x 4 x
0 .5 x 0 .2 4 x
3
3
3
= -2.7x + 77x - 422.22
2
Taking dR(x,4x−33.33)/dx = 0 produces
a Maximum at
x
Chabot College Mathematics
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77
5 .4
14 . 26
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2
Example Production Constraints
SUMMARY:
• Have 3 boundary critical points to consider:
– (0,15), (8 0), and (14.26, 23.71)
• We would normally also consider the only critical
point on the interior (found in the previous
Example 2). However, this point does not satisfy
the condition that there be at least twice as many
health fusion nuts as cashews, so it is omitted.
• We then compare revenue for each of those
three boundary points and identify the largest
revenue.
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Example Production Constraints
SUMMARY:
• We then compare revenue for each of
those three boundary points and identify
the largest revenue
• Tabulating the Results
(x,y)
(0,15)
(8,0)
(14.26, 23.71)
R
45
32
126.76
Pricing the cashews at $14.26 and Trialby-Trail at $23.71 provides maximum
revenue, given the constraints.
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WhiteBoard Work
Problems From §7.3
• P51 → Box Design Optimization
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All Done for Today
Saddle
Point
City
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Chabot Mathematics
Appendix
r s r s r s
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
2a
–
Chabot College Mathematics
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2b
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Chabot College Mathematics
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Chabot College Mathematics
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