Problem $1$: Consider the function $f(x,y)=x^2+5y^2-4xy-(j+1)x-(k+1)y+jk+1.$

$($a$)$ Find the critical point of $f.$ Then show that it is a local minimum using the Second Derivative Test.

$($b$)$ The local minimum you found in part $($a$)$ is actually the global minimum of $f.$ In this part, your

aim is to find the minimum value of $f$ approximately using the gradient descent method. For this purpose,

use the following:

Initial point: $x_0=(\frac{5j+2k+7}{2}+0\mathrm{.}2,\frac{2j+k+3}{2}-0\mathrm{.}2)$

Learning rate: $=0\mathrm{.}1$

$=0\mathrm{.}035$

Stopping criterion: $||gradf||<mi\; id="EditorElement\_572097"></mi>$

You can learn the details of the gradient descent method from the lecture notes or from any online source.

In this problem, you are encouraged to get help from a computer and$/$or a website; you may even write a

small python$/$MATLAB function to carry out the necessary work. In case you use a computer or a web site,

please include the relevant prints$/$screenshots in your homework.

j$=3$ k$=1$