Engineering Calculus Notes 333

# Engineering Calculus Notes 333 - 321 3.6 EXTREMA The...

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Unformatted text preview: 321 3.6. EXTREMA The lynchpin of our strategy for ﬁnding extrema in the case of single-variable 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 single-variable 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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## This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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