12.7 Extreme Values

12.7 Extreme Values - 1 2 .7 .

E x t r e m e 
V a l...

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E x t r e m e 
V a l u e s 
 R 2
 
 
 Local
extrema
occur
at
critical
points
where
 f ′ ( x ) = 0 
or
 f ′ ( x ) 
is
undefined.
 Tests
 1. First
derivative
test
 
 
 
 2. Second
derivative
test
 
 
 
 
 R3 
 
 , Local
extrema
occur
at
critical
points
where
 ∇f ( xy ) = 0 
or
 ∇f ( x, y ) 

is
undefined.



Note:

when
 ∇f ( x, y ) = 0 ,

fx
=
0
 and
fy
=
0
simultaneously.
 
 A
saddle
point
is
a
point
(a,
b,
f(a,
b))
whose
tangent
plane
is
 horizontal
( ∇f ( x, y ) = 0 )
and
f
has
both
higher
and
lower
 function
values
on
any
region
containing
(a,
b).
 
 
 Local
extrema
=
local
maxima,
local
minima,
and
saddle
points.
 
 Second
Partials
Test
 Let
z
=
f(x,
y)
and
suppose
 ∇f ( a, b) = 0 .

Define
D(a,
b),
“the
 discriminant
of
f”,
as
follows:
 D(a,
b)
=
fxx(a,
b)fyy(a,
b)
–
(fxy(a,
b))2
 
 1. If
D(a,
b)
>
0
and
fxx(a,
b)
>
0,
then
f
has
a
local
minimum
 at
(a,
b)
 2. If
D(a,
b)
>
0
and
fxx(a,
b)
<
0,
then
f
has
a
local
maximum
 at
(a,
b)
 3. If
D(a,
b)
<
0,
then
f
has
a
saddle
point
at
(a,
b)
 4. If
D(a,
b)
=
0,
then
it’s
inconclusive
 
 Ex.

f(x,
y)
=
x4
+
y3
+
32x
–
9y
 
 
 
 
 
 
 
 
 Ex.
f(x,
y)
=
xy2
–
6x2
–
3y2
 
 
 
 
 
 
 
 
 
 
 
 
 Do:

Find
all
local
extrema
for
f(x,
y)
=
6x2
‐
2x3
+
3y2
+
6xy
 
 
 
 
 
 
 
 Absolute
Maxima
and
Minima
 If
f
is
a
continuous
function
on
a
closed
and
bounded
region
R,
 then
f
has
an
absolute
maximum
and
absolute
minimum.

The
 absolute
maximum
and
minimum
will
always
be
at
a
critical
 point
or
on
the
boundary.
 
 




 
 
 
 






















 
 
 
 
 
 R2 
 
 
 
 
 
 1.0 0.5 -6 -4 -2 2 4 6 - 0.5 - 1.0 

























 
 
 R3 
 










 
 Steps:
 1. 2. 3. 4. Find
the
critical
points
and
list
them.
 List
the
end
points
or
corners
 Find
the
critical
points
on
the
boundaries
 Find
the
function
values:

the
largest
is
the
absolute
 maximum
and
the
smallest
is
the
absolute
minimum.
 
 Ex.

Find
the
absolute
maximum
and
minimum
for

 f(x,
y)
=
x4
+
y3
+
32x
–
9y
on
the
region
‐2
≤
x
≤
0
and
‐2
≤
y
≤
0
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex.

f(x,
y)
=
xy2
–
6x2
–
3y2
on
the
triangular
region
with
 vertices
(1,
1),
(‐1,
1),
and
(‐1,
‐3).
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex.

f(x,
y)
=
xy2
–
6x2
–
3y2
on
the
region
bounded
by
y
=
x2
and
 y
=
1.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Ex.

f(x,
y)
=
x
+
y2
on
the
region
bounded
by
x2
+
y2
≤
1.
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ...
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This note was uploaded on 10/02/2011 for the course AERO 1234 at Virginia Tech.

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