Problems

The Newton method (see Problem 61328) does not always allow us to approach the root of the equation $f(x)=0$. Find the initial condition $x_0$ for the polynomial $f(x)=x(x-1)(x+1)$ such that $f(x_0)\ne x_0$ and $x_2=x_0$.

Prove that the tangent to the graph of the function $f(x)$, constructed at coordinates $(x_0,f(x_0))$ intersects the $Ox$ axis at the coordinate: $x_0-\frac{f(x_0)}{f^{\prime }(x_0)}$.