$(20$ pts$)($Coding$)$ First use MATLAB to compute the solution, u$,$ of

this problem. Then, implement Newton method to solve the problem,

try different initial points $($parametrized by $\backslash$delta $)$ with u$+\backslash$delta

$1$

$1$

$.$ Draw

the contour plot and trajectories taken by $\backslash$delta $=0\mathrm{.}1,0\mathrm{.}5,1,2,4.$ Supposing

everytime the path getting into the domain u$1<mtext></mtext><mi></mi><mn>5</mn>$ is considered a failed

solution, which $\backslash$delta can be solved successfully?

$(20$ pts$)($Coding$)$ How does Newton method converges? Plot $\backslash |\backslash$epsi $\backslash |2$ vs

number of iterations.