A three$-$spring system can be seen in the figure below. By minimizing the potential

energy determine the position of point $1$ if a force of $(10,8)$ acts on it$.$ Each spring

has a sprint constant of $20$ units.

a$)$ Formulate the optimal design problem. Clearly state the objective function,

design variables, and constraints $($if any$).$

b$)$ Solve the problem analytically. Demonstrate that the point you are obtaining is

indeed the optimal solution.

c$)$ By "hand", solve the problem using steepest descent method for two iterations

starting from the point $x_0={\left[\begin{array}{cc}-0\mathrm{.}2& -0\mathrm{.}6\end{array}\right]}^T.$ Use your Brent's polynomial

approximation code for the line search to help with the calculation of $.$

d$)$ Repeat the previous question but this time use the conjugate gradient method.