Consider the function f $($x$1,$ x$2)=(2$x$11)4+($x$1+$ x$21)2.$

$($a$)$ Find the global minimum of f $,$ and justify your answer.

$($b$)$ Starting at x$(0)=(0,0),$ perform gradient descent with backtracking line$-$

search.

i$.$ Starting at x$(0)=(0,0)$ with stepsize, which is also called learning rate in

the machine learning community, $(0),$ write down the gradient descent

equation for x$(1).$

ii$.$ Suppose we want to set $(0)$ using backtracking line search with $=0\mathrm{.}2$

and Armijo$$s condition f $($x$(1))$ f $($x$(0))$ $(0)$$$f $($x$(0))2$

$2.$ Find a

value of $(0)$ that satisfies this.

iii. Suppose instead you started with $(0)=1$ and an update of $(0)1$

$2$ $(0)$

$($i$.$e$.$ $=1$

$2).$ In the worst case, how many steps of back$-$tracking would

you have to take before accepting x$(1)$