14.7: MAXIMUM AND MINIMUM VALUES
KIAM HEONG KWA
1.
Local Extreme Values and Critical Points
A function
f
of two variables is said to have a local maximum
at a
point (
x
M
,y
M
) if
f
(
x,y
)
≤
f
(
x
M
,y
M
) for all (
x,y
) suﬃciently close to (
x
M
,y
M
)
.
The number
f
(
x
M
,y
M
) is called a local maximum value
of
f
. Similarly,
f
is said to have a local minimum
at a point (
x
m
,y
m
) if
f
(
x,y
)
≥
f
(
x
m
,y
m
) for all (
x,y
) suﬃciently near (
x
m
,y
m
)
.
The number
f
(
x
m
,y
m
) is called a local minimum value
of
f
. Local
maximum values and local minimum values are collectively known as
local extreme values
.
On the other hand, a point (
x
0
,y
0
) in the domain of a function
f
(
x,y
)
is called a critical point
(or stationary point) of
f
•
if
f
x
(
x
0
,y
0
) =
f
y
(
x
0
,y
0
) = 0, i.e.,
∇
f
(
x
0
,y
0
) =
0
, or
•
if one of these partial derivatives does not exist.
Theorem 1.
If a function
f
(
x,y
)
has a local extreme value at a point
(
x
0
,y
0
)
, then the point
(
x
0
,y
0
)
must be a critical point of
f
.
Proof.
If
f
x
(
x
0
,y
0
) does not exist, then there is nothing to prove. Sup
pose
f
x
(
x
0
,y
0
) exists. Deﬁne
g
(
x
) =
f
(
x,y
0
) for all
x
suﬃciently
close to
x
0
. Then
g
(
x
0
) must be a local extreme value of
g
since
f
(
x
0
,y
0
) is a local extreme value of
f
, from which it follows that
f
x
(
x
0
,y
0
) =
g
0
(
x
0
) = 0. Likewise, either
f
y
(
x
0
,y
0
) does not exist or
one can show that
f
y
(
x
0
,y
0
) = 0.
±
Remark 1.
The idea used in the proof follows from the geometrical
observation that if
(
x
0
,y
0
,f
(
x
0
,y
0
))
is the highest point in one of its
suﬃciently small neighborhood, then it is also the highest point along
the curve
r
(
x
) =
< x,y
0
,f
(
x,y
0
)
>
in the neighborhood.
Date
: September 30, 2010.
1
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KIAM HEONG KWA
Corollary 1.
If
f
(
x,y
)
is a diﬀerentiable function and has a local
extreme value at a point
(
x
0
,y
0
)
, then all its directional derivatives
vanish at
(
x
0
,y
0
)
.
Proof.
Since
f
is diﬀerentiable,
∇
f
(
x
0
,y
0
) exist. Furthermore, for any
unit vector
u
,
D
u
f
(
x
0
,y
0
) =
∇
f
(
x
0
,y
0
)
·
u
. The conclusion now follows
from the previous theorem.
±
Second Derivatives Test.
Let
f
have continuous second partial deriva
tives near a critical point (
x
0
,y
0
). In particular,
f
x
(
x
0
,y
0
) =
f
y
(
x
0
,y
0
) =
0. The Hessian matrix
of
f
at (
x
0
,y
0
) is the matrix
±
f
xx
(
x
0
,y
0
)
f
xy
(
x
0
,y
0
)
f
yx
(
x
0
,y
0
)
f
yy
(
x
0
,y
0
)
²
.
Note that
f
xy
(
x
0
,y
0
) =
f
yx
(
x
0
,y
0
) by the Clairaut’s theorem in section
14.3. The discriminant
of
f
at (
x
0
,y
0
) is the determinant of the Hessian
matrix, i.e.,
D
(
x
0
,y
0
) =
³
³
³
³
f
xx
(
x
0
,y
0
)
f
xy
(
x
0
,y
0
)
f
yx
(
x
0
,y
0
)
f
yy
(
x
0
,y
0
)
³
³
³
³
(1.1)
=
f
xx
(
x
0
,y
0
)
f
yy
(
x
0
,y
0
)

[
f
xy
(
x
0
,y
0
)]
2
.
•
Suppose
D
(
x
0
,y
0
)
>
0.
–
If
f
xx
(
x
0
,y
0
)
<
0 or
f
yy
(
x
0
,y
0
)
<
0, then
f
(
x
0
,y
0
) is a local
maximum.
–
If
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 Fall '10
 Kwa
 Calculus, Geometry, Critical Point, x0

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