 Solution

At $x=1$, the derivative and curvature are $f^{\prime }(1)=8$, $f^{\prime \prime }(1)=-8$, so the next iterate is $2$. At $x=2$, they are $f^{\prime }(2)=4$, $f^{\prime \prime }(2)=4$, so the next iterate is $1$. HenceThe undamped algorithm does not converge to a minimum; it has an exact two-cycle. The implementation in (iii) detects this after returning to one and reports failure. This example illustrates why the local convergence result in (ii) does not justify global success.