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Unformatted text preview: ) < g ( x ), i.e. that h := fg is negative on ( s,t ) (note h ( s ) = h ( t ) = 0). If f 00 ( x ) > 0 for a < x < b , then h is also, since g 00 = 0. Suppose h were not negative on ( s,t ); then it would have a local max on this interval at some point c . By Part (1), this would imply h 00 ( c ) ≤ 0. < . Part (i) holds, in particular, for g ( t ) = f ( x ) + f ( x )( xt ). 5.4.2 Taylor’s Thm Deﬁnition 5.4.3. If f ∈ C n , the Taylor expansion of f at a is T n ( a,x ) = f ( a ) + f ( a )( xa ) + 1 2 f 00 ( a )( xa ) 2 + ··· + 1 n ! f ( n ) ( a )( xa ) n , where f ( n ) = ( f ( n1) ) is deﬁned by induction. Theorem 5.4.4. If f ( n ) exists at a , then T n ( a,x ) is the unique polynomial of degree n in powers of ( xa ) having n thorder agreement with f ( x ) at a ....
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 Fall '11
 Wong
 Continuity, Derivative, Tn, local max, strict local min, Inverse Fn Thm

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