From part$($a$),$ X$=[2,-2]$ is the minimum point for f$1($x$)))$ Consider the following unconstrained multi$-$objective optimization problem with three objective functions all of which are to be minimized :

design variables $=\{[$ x$\_1$; x$\_2$; x$\_3$; x$\_4]\}$

minimize f$\_1()=$x$\_1^3-2$ x$\_1^2+$x$\_2^2+2$ x$\_1$ x$\_2+5$

f$\_2()=$x$\_2^2+$x$\_3^2-5$ x$\_2$ x$\_3$

f$\_3()=2$ x$\_4^2-$x$\_1^2-2$ x$\_3^2+4$ x$\_1$ x$\_3-4$ x$\_3$ x$\_4$

Note that f$\_1()$ is the same function as the objective function in part $($a$).$ The most important objective is f$\_1,$ and the second most important is f$\_2,$ and the third most important is f$\_3.$ With the help of your answer from part $($a$),$ apply lexicographic optimization to determine the minimum points for this multi$-$objective problem.