Engineering Calculus Notes 333

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

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.

Ask a homework question - tutors are online