Answer this T/F question!!

Question

If  has a root, then Newton's Method starting at  will approximate the root nearest .

Answer 1

The easiest way to disprove this is with a counterexample. Consider f(x) = x * sqrt(1-x)

f'(x) = sqrt(1-x) - x/(2(sqrt(1-x)))

Roots are at 0 and 1.

Start at .6, which is closer to 1.

xn+1 = xn - f(xn)/f'(xn)

We converge to the solution at 0