Answer Key
Maxima and Minima of Functions: Critical Points and Extrema
-
University:
Amherst College -
Course:
MATH 111 | Introduction to the Calculus Academic year:
2024
-
Views:
71
Pages:
4
Author:
Teagan Allison
Maxima and Minima of Functions of Several Variables
( from Stewart , Calculus, Chapter 14)
Critical Points and Local Extrema
JA
Critical Point: A point ( a , 6) in the domain of / where
fx ( a , b ) = 0 and fv ( a, b ) = 0
or where one of these partial derivatives does not exist.
local
maximum
y
Thm. If / has a local maximum or local minimum at ( a, 6) then
( a , b ) is a critical point of /.
Is
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T
Sv
Second Derivative Test Suppose the second partial derivatives of / are continuous near ( a, b ) and suppose
that fx ( a , b ) = 0 and fy ( a , b ) = 0 ( that is, (a , b ) is a critical point of / ) Let
f
{** (/«)(/ ) -(A )2
D( x, y )
Jyy
x
y
J
‘
be the determinant of the Hessian matrix of second order partial derivatives of /.
j (x
- w "? 0 =*> foorave 'A f
v (a) If D( a , b ) > Q and fXx ( a , b ) > 0, then / ( a , b ) is a local minimum. |
( b) If D ( a , b ) > 0 and fxx (.a , b ) < 0, then / (a , b ) is a local maximum. P/x 6 ? CcMayf
(c) If D( a , 6) < 0, then fa , b ) is a saddle point.
(d ) If D = 0, the test gives no information - / could have a local maximum or local minimum at (a, b ) , or
( a , b ) could be a saddle point.
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.
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minimum
absolute \
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,
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absolute
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Ex. 1 Let f { x , y ) a:4 + 2 y 2 - 4 xy. Find all critical points of /, and classify each as a local maximum ,
local minimum , or saddle point.
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-' A Method of Lagrange Multipliers
To find the maximum and minimum values of a function / subject to the constraint g = k for some constant
k ( assuming these extreme values exist and Vg 0 when g = k ):
^
1. Find all points where
g=k
and
V / = A Vs
for some A 6 R . These values of A are called the Lagrange multipliers.
2. Evaluate / at each of the points from Step 1. The largest is the maximum value of
the minimum value of / , subject to the constraint .
/ and the smallest is
Ex. 2. Find the absolute maximum and minimum values of f ( x , y ) = x2 + 2y2 on the circle x 2 + y2 * 1.
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Absolute Maxima and Minima
If f ( x , y ) is continuous on a closed ,
Extreme Value Theorem for Functions of Two Variables
2
bounded set D in R , then / attains an absolute maximum value / (11, 2/1 ) and an absolute minimum value
/ jx2, 2/2 ) at some points (21, 1/1) and (12, 2/2) in D. /These extreme values occur either at the critical points
of / or on the boundary of D
.
.
.
To find
( 1)
(2)
(3)
~
the absolute maximum and minimum values of a continuous function / on a closed, bounded set D :
Find the values of / at each critical point in D.
Find the maximum and minimum of f ( x , y ) on the boundary of D.
The largest of these values is the absolute maximum and the smallest is the absolute minimum.
—
Ex. 3. Find the points at which the absolute maximum arid minimum values of the function f { x , y )
2x2 + 3i/2 - A x - 5 on the disk x2 + y2 < 16 occur. State all points where the extrema occur as well as the
maximum and minimum values.
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Ex. 5 A rectangular box without a lid is to be made from 12 m 2 of cardboard Find the maximum volume
of such a box.
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