Find the minimum of the following function using optmality conditions. Then, bracket the minimum $(xin[a,b])$ by starting from $x^{(0)}=0$ and using bounding phase method with $=0\mathrm{.}5.$ Use an excel sheet to apply the proceedings of the algorithm. Submit your worked$-$out excel sheet. Show that the optimal solution is included within the brackets obtained.

$f(x)=x^3-10x^2+10$

For the above problem and bounds $([a,b])$ obtained, use three iterations of the Golden Section search method to reduce the search space. Use an excel sheet to apply the proceedings of the algorithm. Submit your worked$-$out excel sheet. Show that the optimal solution is always included in the reduced bounds at every iteration.

We need to find the minimum of the following function:

$f(x_1,x_2)=2x_1^4+x_2^4-28x_1^2{x}_2-10x_1+2500,5(x_1,x_2)10.$

$($a$)$ Find the minimal solution using the optimality conditions by trial and error.

$($b$)$ In order to simulate the steepest descent method for minimization, we shall use an excel spreadsheet $($rows and columns$)$ as our search space. The column represents $x_1$ and is marked values from $5\mathrm{.}0($first row$)$ to $10\mathrm{.}0(26$th row$)$ in steps of $0\mathrm{.}2.$ Similarly, rows represent $x_2$ with $26$ entries marking values from $5\mathrm{.}0($first column$)$ to $10\mathrm{.}0(26$th column$)$ in steps of $0\mathrm{.}2.$ Thus, the search space has $2626$ points in total. We start with $x^{(0)}=(8\mathrm{.}6,9\mathrm{.}6)$ grid on the worksheet. At $x^{(0)},$ we consider eight neighboring grids $($in all directions$)$ and locate the grid having the smallest $f$ value. Then, move from $x^{(0)}$ in the direction of the neighboring smallest valued grid until the function value starts to increase. Stop at this grid and declare it as $x^{(1)}$ point. Continue this process until you end up at a grid with no better neighboring grid. Is this the optimal point you found in Item $1?$ Submit a screenshot of the excel sheet by marking teh starting grid and every other intermediate grids until the final grid. Join the consecutive grids with arrows to show the progress of the search.

$($c$)$ Contrast and compare the above simulated search procedure with the true steepest descent method of minimization in terms of their scope and accuracy.