CM221A
ANALYSIS I
NOTES ON WEEK 9
FINDING MAXIMAL AND MINIMAL VALUES
Deﬁnition.
Let
f
be a function deﬁned on an interval (
a,b
). We say that
f
has a
local maximum at a point
c
∈
(
a,b
) if there exists
ε >
0 such that
f
(
c
)
>
f
(
x
) for
all
x
∈
(
c

ε,c
+
ε
). Similarly,
f
has a local minimum at
c
∈
(
a,b
) if there exists
ε >
0 such that
f
(
c
)
6
f
(
x
) for all
x
∈
(
c

ε,c
+
ε
).
Theorem.
Let
f
be a diﬀerentiable function on the interval (
a,b
). If
f
has a local
maximum or a local minimum at
c
∈
(
a,b
) then
f
0
(
c
) = 0.
Proof.
If
f
has a local maximum at
c
then there exists
ε >
0 such that
f
(
x
)

f
(
c
)
x

c
>
0
for all
x
∈
(
c

ε,c
) and
f
(
x
)

f
(
c
)
x

c
6
0 for all
x
∈
(
c,c
+
ε
). Therefore, in the
deﬁnition of the derivative, the left limit is nonnegative and the right limit is
nonpositive. Since
f
is diﬀerentiable, both these limits exist and coincide. This
implies that they are equal to zero. The corresponding result for a local minimum is
obtained in a similar way (or by applying the local maximum result to the function
g
(
x
) =

f
(
x
)).
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 Spring '09
 Calculus, Rolle, local maximum, Rolle's theorem

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