$(30$ marks$)$ The Potential Energy P$($x$1,$ x$2)$ of a dynamic problem, which consists of two wheeled carts connected to each other is given by the equation below. P$($x$1,$x$2)=5.2$x$12+1.8$x$222$ x$1$x$2$ x$12$x$2$ In order to determine the displacements $($x$1,$ x$2)$ of the carts, the potential energy needs to be minimised. Minimize this function using the Nelder$-$Mead simplex method, starting with x $0=[$x$1$ x$2]=[7.05.0],$ an initial step size c$=2,\backslash$alpha $=1.0,\backslash$beta $=0.5,\backslash$gamma $=2.0,\backslash$rho $=0.5$ and tol $=0\mathrm{.}001.$ Perform the Nelder$\_$Mead search until the convergence criterion $($f$($xL$)$ f$($xCE$))2+($f$($xM$)$ f$($xCE$))2+($f$($xH$)$ f$($xCE$))23<$ tol is satisfied.