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Real Analysis: Derivatives and Sequences

Suppose that f: [a,b] &#61664; R is differentiable, that 0 < m f '(x) M for x &#1108; [a,b], and that f(a) < 0 < f(b). Show that the equation f(x) = 0 has a unique root in [a,b]. Show also that for any given x1 &#1108; [a,b], the sequence (xn), xn+1 = xn - for n = 1, 2,..., is well defined (i.e. for each n, xn &#1108; [a,b]), and that xn &#61664; as n &#61664; &#8734;, where &#1108; [a,b] is such that f( ) =0.

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Suppose that f: [a,b]  R is differentiable, that 0 < m f '(x) M for x є [a,b], and that f(a) < 0 < f(b). Show that the equation f(x) = 0 has a unique root in [a,b]. Show also that for any given x1 є [a,b], the sequence (xn), xn+1 = xn - for n = 1, 2,..., is well defined (i.e. for each n, xn є [a,b]), and that xn  as n  ∞, where є [a,b] is such that f( ) =0.

Proof. (I) To prove that the equation f(x) = 0 has a unique root in [a,b]. As f(a) < 0 < f(b) and f: [a,b]  R is differentiable, we ...

Solution Summary

Derivatives and sequences are investigated. Unique roots of functions are differentiated.

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