Clear[x1,x2,x3]

Solve[{0+lambda1(-1)+lambda2(2x1)==0,0+lambda1(-1)+lambda2(2x2)==0,2x3+lambda1(-2)+lambda2(-1)==0,-x1-x2-2x3+2==0,x1^2+x2^2-x3==0},{x1,x2,x3,lambda1,lambda2}]

x1=1/2
x2=1/2
x3=1/2

rp1=RegionPlot3D[-(y1-x1)-(y2-x2)-2(y3-x3)≤0,{y1,-3,3},{y2,-3,3},{y3,-1,3},Mesh->None,LabelStyle->(FontSize->14),PlotPoints->20,PlotStyle->Directive[GrayLevel[1/10,1/10]],BoundaryStyle->None]

rp2=RegionPlot3D[(y1-x1)+(y2-x2)-(y3-x3)≤0,{y1,-3,3},{y2,-3,3},{y3,-1,3},Mesh->None,LabelStyle->(FontSize->14),PlotPoints->20,PlotStyle->Directive[GrayLevel[1/10,1/10]],BoundaryStyle->None]

feasible=RegionPlot3D[-y1-y2-2y3≤-2 && y1^2+y2^2-y3≤0,{y1,-3,3},{y2,-3,3},{y3,-1,3},Mesh->None,LabelStyle->(FontSize->14),PlotPoints->100,PlotStyle->Directive[GrayLevel[4/10,4/10]],BoundaryStyle->None]

Show[feasible,rp1,rp2]