(8/18/08)
Math 20C. Lecture Examples.
Section 14.7, Part 1. Maxima and minima: The firstderivative test
†
Definition 1
(a)
A function
f
(
P
)
of two or three variables has a
global maximum
at a point
P
0
if
f
(
P
)
≤
f
(
P
0
)
for all points
P
in its domain. It has a
global minimum
at
P
0
if
f
(
P
)
≥
f
(
P
0
)
for all
P
in its domain.
(b)
The function has a
local maximum
at
P
0
if it is defined in an open circle (in the twovariable
case) or a open sphere (in the threevariable case) centered at
P
0
and
f
(
P
)
≤
f
(
P
0
)
for all points
P
in
the circle or sphere . The function has a
local minimum
at
P
0
if
f
(
P
)
≥
f
(
P
0
)
for all points
P
in such
a circle or sphere.
Recall that
x
=
a
is a critical point of a function
y
=
f
(
x
) of one variable if the function is defined
in an open interval containing
a
and if either
f
prime
(
a
) is zero or
f
prime
(
a
) does not exist. The definition of critical
points for functions of two or three variables is similar.
Definition 2
(a)
The point
P
(
a, b
)
is a
critical point
of
z
=
f
(
x, y
)
if the function is defined in an
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 Spring '08
 Helton
 Calculus, Derivative, Web page, 1 foot, 2 dollars, 2 feet, 1 dollar

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