Let$$s suppose we have a function representing a surface, namely

z $=$ x$24$x $+$ y$412$y$3+54$y$2108$y $+85(8)$

We don$$t know where the minimum $($best solution$)$ might be$.$ We guess x $=0$ and y $=0$

because this takes zero effort.

$($a$,2$ pts$)$ Compute the two$-$dimensional negative gradient, $$~$($z$)=($z$/$x$,$z$/$y$)$

and evaluate it at $(0,0).$ In which angular direction is it pointing?

$($b$,2$ pts$)$ Algorithms need a $$step length$,$ $,$ to move along the negative gradient. Most

of the time this is a value that starts at $=1.$ Take a $$unit step$,$~$($z$)$ in the direction

of the negative gradient. If the new value of z at the updated position $($x$1,$ y$1)$ is lower

than the value at $(0,0)$ then we can accept the updated value. If not, we can divide by

two and try that. We repeat until we get a lower value. Report your new coordinates

$($x$1,$ y$1)$ and your first successful step length $.$

$($c$,2$ pts$)$ Repeat what you did in $($b$)$ except update the starting value of to twice its

value the last time it worked. If the new try is not lower than the current lowest value

of z then divide the step$-$length by $2.$ Note that you have to use the updated position

$($x$1,$ y$1)$ to get the new direction of descent. Repeat until you are successful. What is

the first successful try you make, and what is the new value of z at the updated coordi$-$

nates $($x$2,$ y$2)?$ Please give the step length, $,$ the values of x$2,$ y$2$ and the function value z$.$