Someone tries to find a root of using Newton-Raphson

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Computational Physics, Ch Where a function vanishes, and what lies between the points

Problem

Someone tries to find a root of f(x)=x3−3x+1=0f(x) = x^3 - 3x + 1 = 0 using Newton-Raphson, starting from x0=1x_0 = 1. They write: "Newton's step is x1=x0−f(x0)/f′(x0)x_1 = x_0 - f(x_0)/f'(x_0). Here f(1)=1−3+1=−1f(1) = 1 - 3 + 1 = -1 and f′(1)=3(1)2−3=0f'(1) = 3(1)^2 - 3 = 0. Since f′(1)=0f'(1) = 0, no update can be found, so the tangent line at x=1x=1 is horizontal and the iteration cannot proceed. I conclude that the equation has no solution near x=1x=1, and since Newton's method is the standard technique for equations that cannot be rearranged, the equation must have no real root at all near there." ((a)) Identify the error in this reasoning. ((b)) Using values of ff at a few integers or simple fractions on either side of x=1x=1, bracket what is actually true about the roots of ff near x=1x=1, and give each root to three significant figures by narrowing its bracket.

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