Problem $2(30$ point$)$ Consider f$($x$1,$ x$2)=?0\mathrm{.}5$x$1$x$2+0\mathrm{.}2$ cos$($x$1)0\mathrm{.}2$x$220\mathrm{.}5$x$2+0\mathrm{.}1?($i$)10$ point Show that the above function has a unique fixed point on $[0,1]\backslash$times $[0,1].$ Also, show that the fixed point iteration converges to the fixed point of f$.$ You need to use the theorems discussed in class. $($ii$)10$ point Construct the MATLAB code to find the fixed point via the fixed$-$point iteration $($Picard iteration$).($iii$)10$ point Iterative your algorithm for two times when the initial condition is $?0\mathrm{.}10\mathrm{.}2?.$ Note that $?$x$1$ x$2?=?0\mathrm{.}2020\mathrm{.}06?$ is the fixed point of the above equation. Find the absolute and relative errors between your iteration and $?$x$1$ x$2?=?0\mathrm{.}2020\mathrm{.}06?.$ Solution: $($i$)$ the function does not hold f : $[0,1]\backslash$times $[0,1]->[0,1]\backslash$times $[0,1]$ In fact, when f$(0,0)=?0\mathrm{.}20\mathrm{.}2?.$ The differentiation of f holds $|$f$($x$1,$ x$2)|<1,($x$1,$ x$2)\backslash$times $[0,1]\backslash$times $[0,1]$ The initial condition holds x$0$ in $[0,1]\backslash$times $[0,1].($ii$)$ MATLAB codes $($iii$)$ simple computation