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Engineering Calculus Notes 331

# Engineering Calculus Notes 331 - 3.6 EXTREMA 319 Theorem...

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3.6. EXTREMA 319 Theorem 3.6.6 (Extreme Value Theorem) . If S R 3 is compact, then every real-valued function f which is continuous on S achieves its minimum and maximum on S . Note that this result includes the Extreme Value Theorem for functions of one variable, since closed intervals are compact, but even in the single variable setting, it applies to functions continuous on sets more general than intervals. Proof. The strategy of this proof is: first, we show that f must be bounded on S , and second, we prove that there exists a point s S where f ( s ) = sup x S f ( x ) ( resp . f ( s ) = inf x S f ( x )). 12 Step 1: f ( x ) is bounded on S : Suppose f ( x ) is not bounded on S : this means that there exist points in S at which | f ( x ) | is arbitrarily high: thus we can pick a sequence s k S with | f ( s k ) | >k . Since S is (sequentially) compact, we can find a subsequence—which without loss of generality can be assumed to be the whole sequence—which converges to a point of S : s k s 0 S . Since f ( x ) is continuous on S , we must have f ( s k )
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