Finding absolute extrema for “nice” functions
Math 8 2009, Chris Lyons
Let’s suppose you’re given a function
f
: [
a, b
]
→
R
and you know that it’s continuous. Even this little
bit of information already tells you something:
Theorem 1
(Thm 3.12 in Apostol)
.
If
g
: [
α, β
]
→
R
is continuous, then there is at least one point
γ
∈
[
α, β
]
such that
g
(
γ
)
is the absolute maximum of
g
. Similarly, there is at least one point
δ
∈
[
α, β
]
such that
g
(
δ
)
is the absolute minimum of
g
.
So by this theorem you know there are
x
values in [
a, b
] where
f
takes its absolute maximum and absolute
minimum... but how do you actually
find
these
x
values? If you look at the proof of Theorem 1, it’s of no
help: it shows you that these
x
values exist, by using the completeness axiom for the real numbers, but it
doesn’t actually give a way of finding them. (Proofs like this are called “nonconstructive” proofs.)
But if your function
f
is nice enough, then you can actually find these points! We’ll see at the end what
I mean by “nice enough”. First, let’s make the following observation, which may look kind of useless, but is
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 Fall '08
 Vuletic,M
 Math, Calculus, relative maximum, Apostol, absolute maximum, absolute minimum, final conclusion

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