Consider a scalar function $F \in C^{4}$ with a root $x^{\star) $ of multiplicity two, i.e., $$ F\left(x^{\star} ight)=F^{\prime} \left(x^{\star} ight)=0, \quad F^{\prime \prime}\left(x^{\star} ight) ego $$ Find a natural number $m$ so that the modified Newton method $x_{k=1}=\Phi\left(x_{k} ight) with Phi(x)=x-m \frac{F(x)}^{\prime} (x)} $$ converges at least quadratically. Tip: make use of the statement from Q1.3 on the second assignment (don't worry about the smoothness of $\Phis. Also recalling L'Hopital might prove helpful. Answer: SP.SD.405 $$