Exercise $5\mathrm{.}2\mathrm{.}4$ Write a program to find the global minimum value of $f=(x-1{)}^4-x+(y-2{)}^2$ using

steepest decent and plot the progress of the algorithm. Use the initial point $(1,1)$ and a step size of

$s=0\mathrm{.}1.$ Also track the $(x,y)$ values during the execution of the algorithm, using a scatter plot. For

the assignment, submit the code and the plots.

Exercise $5\mathrm{.}2\mathrm{.}5$ In this problem you will use the steepest decent algorithm to try to find the global

minimum value of the function $f(x,y)=x^2(x-2)(x+3)+y^2(y-2)(y+3),$ which is not convex.

Modify the Python code in the text accordingly for each of the parts below. Play with the step size

until you get smooth plots. For the assignment, submit the plots for each part. Submit the code listing

only for part $($a$).$

$($a$)$ Use $(1,1)$ as the initial point. Plot $f(x,y)$ against the number of iterations.

$($b$)$ Repeat with $(1,-1)$ as the initial point.

$w_{11}$:$=w_{11}-\frac{del{f}_e}{del{w}_{11}}$

Exercise $6\mathrm{.}4\mathrm{.}1$ Derive the back propagation update rule for $w_{12}$ and $w_{23}$ in the $3/2-1-$neural network

above, by following the same steps as above. Start with a step similar to the one in $(6\mathrm{.}44)$ and applying

the chain rule.

Exercise $6\mathrm{.}4\mathrm{.}2$ Derive the back propagation up@rules for $ww{w}_2,w{w}_{21},$ and $w_{12},$ in the $2/3-2-1-$

neural network above, by following the same steps as above.

Exercise $6\mathrm{.}5\mathrm{.}1$ Derive the back propagation update rule for $w{w}_{21}$ and $w_{13}$ in the $3/3-2-1-$neural net$-$

Owork with activating functions above, by following the same steps as above.