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A root $\alpha$ of equation $f(x)=0$ can be computed to any degree of accuracy if a 'good' initial approximation $x_0$ is chosen for which

  1. $f(x_0) > 0$
  2. $f (x_0) f''(x_0) > 0$
  3. $f(x_0) f'' (x_0) < 0$
  4. $f''(x_0) >0$

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F(x)>0

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