Engineering Calculus Notes 337

Engineering Calculus Notes 337 - 3.6. EXTREMA 325 and the...

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Unformatted text preview: 3.6. EXTREMA 325 and the value there is g parenleftbigg 1 2 parenrightbigg = f parenleftbigg 1 2 , 1 2 parenrightbigg = − 1 2 . But we also need to consider what happens when bardbl ( x,y ) bardbl → ∞ in our set. It is easy to see that for any point ( x,y ), f ( x,y ) ≥ x 2 − 2 x ≥ − 1, and also that x 2 − 2 x → ∞ if | x | → ∞ . For any sequence ( x j ,y j ) with bardbl ( x j ,y j ) bardbl → ∞ , either | x | → ∞ (so f ( x,y ) ≥ x 2 − 2 x → ∞ ) or | y | → ∞ (so f ( x,y ) ≥ y 2 − 1 → ∞ ); in either case, f ( x j ,y j ) → ∞ . Since there exist such sequences with x j ≤ y j , the function is not bounded above. Now, if −→ s i = ( x i ,y i ) is a sequence with x i ≤ y i and f ( −→ s i ) → inf x ≤ y f ( x,y ), either −→ s i have no convergent subsequence, and hence bardbl −→ s i bardbl → ∞ , or some accumulation point of −→ s i is a local minimum for f . The first case is impossible, since we already know that then...
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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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