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An equation f(x)=0 has 2 roots in the interval between 0 and 8. What is the value or a description for which the Newton Raphson method will fail to converge?

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You haven't given enough information for there to be anything more than a tautological answer to your question. Do you know anything about $f$, other than it having 2 roots? If not, there's not much to say. – Ryan Budney Jan 6 '11 at 6:09
This depends largely on the shape of the graph, and on the choice of starting point, $x_0$. In particular if their are horizontal tangents or vertical asymptotes in the interval [0,8], Newton's method may fail. – user3180 Jan 6 '11 at 6:46
In general, this is a very difficult question. See for instance. – lhf Feb 5 '11 at 10:42

See wikipedia. Since you're asking about a function with a multiple root, you probably want the answer in section 3.

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