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Engineering Calculus Notes 329

# Engineering Calculus Notes 329 - real-valued functions...

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3.6. EXTREMA 317 1. f is bounded (resp. bounded below , bounded above ) on S if f ( S ) is bounded (resp. bounded below, bounded above). 2. The supremum (resp. infimum ) of f on S is defined by sup x S f ( x ) = sup f ( S ) inf x S f ( x ) = inf f ( S ) . 3. The function f achieves its maximum (resp. achieves its minimum ) on S at x S if f ( x ) (resp. ) f ( s ) for all s S. We shall say that x is an extreme point of f ( x ) on S if f ( x ) achieves its maximum or minimum on S at x ; the value f ( x ) will be referred to as an extreme value of f ( x ) on S . In all the statements above, when the set S is not mentioned explicitly, it is understood to be the whole domain of f . The Extreme Value Theorem A basic result in single-variable calculus is the Extreme Value Theorem, which says that a continuous function achieves its maximum and minimum on any closed, bounded interval [ a,b ]. We wish to extend this to result to
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Unformatted text preview: real-valued functions deFned on subsets of R 3 . ±irst, we need to set up some terminology. Defnition 3.6.3. A set S ⊂ R 3 of points in R 3 is closed if for any convergent sequence s i of points in S , the limit also belongs to S : s i → L and s i ∈ S for all i ⇒ L ∈ S. It is an easy exercise (Exercise 9 ) to show that each of the following are examples of closed sets: 1. closed intervals [ a,b ] in R , as well as half-closed intervals of the form [ a, ∞ ) or ( −∞ ,b ]; 2. level sets L ( g,c ) of a continuous function g , as well as sets deFned by weak inequalities like { x ∈ R 3 | g ( x ) ≤ c } or { x ∈ R 3 | g ( x ) ≥ c } ;...
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