Unformatted text preview: 321 3.6. EXTREMA The lynchpin of our strategy for ﬁnding extrema in the case of
singlevariable functions was that every local extremum is a critical point,
and in most cases there are only ﬁnitely many of these. The analogue for
our present situation is the following.
Theorem 3.6.10 (Critical Point Theorem). If f : R3 → R has a local
→→
extremum at − = − 0 and is diﬀerentiable there, then it is a critical point
x
x
→) :
−
of f ( x
→→
−−
→
−
∇ f ( x 0) = 0 .
→
−→
Proof. If ∇ f (− 0 ) is not the zero vector, then some partial derivative, say
x
∂f
→
−
∂xj , is nonzero. But this means that along the line through x 0 parallel to
the xj axis, the function is locally monotone:
d
∂f −
→
→
[f (− 0 + t− j )] =
x
e
(→0 ) = 0
x
dt
∂xj
means that there are nearby points where the function exceeds, and others
→
→
where it is less than, the value at − 0 ; therefore − 0 is not a local extreme
x
x
→
− ).
point of f ( x Finding Extrema
Putting all this together, we can formulate a strategy for ﬁnding the
extreme values of a function on a subset of R3 , analogous to the strategy
used in singlevariable calculus:
→
Given a function f (− ) deﬁned on the set S ⊂ R3 , search for extreme
x
values as follows:
→
1. Critical Points: Locate all the critical points of f (− ) interior to S ,
x
→
− ) at each.
and evaluate f ( x
2. Boundary Behavior: Find the maximum and minimum values of
→
f (− ) on the boundary ∂S ; if the set is unbounded, study the
x
→
limiting values as − → ∞ in S .
x
3. Comparison: Compare these values: the lowest (resp. highest) of
all the values is the inﬁmum (resp. supremum), and if the point at
which it is achieved lies in S , it is the minimum (resp. maximum)
value of f on S .
In practice, this strategy is usually applied to sets of the form
→
→
S = {− ∈ R3  g(− ) ≤ c}. We consider a few examples.
x
x ...
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 Fall '08
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 Calculus, Critical Point, extremum, singlevariable functions

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