(*Exercise 1*)
reg=RegionPlot[x11 (1-x11)>=x12^2,{x11,-1,1},{x12,-1,1}]
Minimize[{2x11+2x12,0<=x11<=1&&x11 (1-x11)>=x12^2},{x11,x12}]//Simplify

opt=ListPlot[{{(1/4) (2-(Sqrt[2])),-(1/(2 (Sqrt[2])))}},PlotStyle->{Red,Thick}];
Show[reg,opt]

Minimize[{2x11+2x12,x11+x22==1&& x11>=0&&x22>=0&&x11 x22-x12^2>=0},{x11,x12,x22}]//Simplify

(*Exercise 2*)
RegionPlot3D[0<=z<=1&&(x-2z)^2+y^2<=z^2,{x,0,3},{y,-1.5,1.5},{z,0,1},PlotPoints->100,BoxRatios->Automatic]

Reduce[Exists[z,0<=z<=1&&(x-2z)^2+y^2<=z^2],{x,y}]

RegionPlot[(x==0&&y==0)||(0<x≤(3/2)&&-((Sqrt[(x^2)])/(Sqrt[3]))≤y≤((Sqrt[(x^2)])/(Sqrt[3])))||((3/2)<x<3&&-(Sqrt[-3+4 x-(x^2)])≤y≤(Sqrt[-3+4 x-(x^2)]))||(x==3&&y==0),{x,0,3},{y,-1.5,1.5},PlotPoints->100]

(*Exercise 2*)

feasible=RegionPlot[x^4+y^4<=1,{x,-1,1},{y,-1,1},PlotPoints->100]

Solve[2x^4==1,x]

cx=1;cy=1;
lines=Plot[Table[c/cy-cx/cy x,{c,-2,2,1/5}],{x,-1,1},PlotStyle->Black];
min=Plot[c/cy-cx/cy x/.c->-(2/(2^(1/4))),{x,-1,1},PlotStyle->Red];
max=Plot[c/cy-cx/cy x/.c->(2/(2^(1/4))),{x,-1,1},PlotStyle->Red];
Show[feasible,lines,min,max]

RegionPlot[x^6+y^6<=1,{x,-1,1},{y,-1,1},PlotPoints->100]

RegionPlot[x^20+y^20<=1,{x,-1,1},{y,-1,1},PlotPoints->100]