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numerical analysis5 - c ∈ a,b such that f c = K...

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2 1. MATHEMATICAL PRELIMINARIES AND ERROR ANALYSIS Theorem (Extreme Value Theorem) . If f C [ a, b ] , then c 1 , c 2 [ a, b ] exist with f ( c 1 ) f ( x ) f ( c 2 ) for all x [ a, b ] . In addition, if f is di ff erentiable on ( a, b ) , then c 1 and c 2 occur either at the endpoints of [ a, b ] or where f 0 is zero. Theorem (Intermediate Value Theorem) . If f C [ a, b ] and K is any number between f ( a ) and f ( b ) , then there exists
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Unformatted text preview: c ∈ ( a,b ) such that f ( c ) = K . Corollary (Intermediate Value Theorem) . If f ∈ C [ a,b ] and f ( a ) f ( b ) < , then there exists c ∈ ( a,b ) such that f ( c ) = 0 . Note. This corollary is often used in approximating roots of equations....
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