A Fractal Image:


This is an picture of the basins of attraction to the roots when Newton's method is applied to the equation x^3 - 1 = 0 in the complex plain. The center of the picture is the origin and the real axis, i.e,. the x-axis runs from -2 to 2 in the picture. The roots are 1 and -1/2 + i3^(1/2)/2 and -1/2 - i3^(1/2)/2. If you start with any point in the blue area and apply Newton's method, the iterates will converge to 1.

From the picture you can see that most of the negative real axis is blue except for a few points like the origin. See if you can find the coordinates of the first point on the negative real axis to the left the origin not in the blue area. In other words find the largest x < 0 whose iterates do not converge to 1.

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