RegionPlot[x +y≤5 && 2x ^2-y≤0 && x≥ 0&& y≥ 0,{x,0,1.5},{y,0,6}]

ContourPlot[2x^2+2y^2-2x y -4 x - 6 y,{x,0,1.5},{y,0,6}]

ContourPlot[2x^2+2y^2-2x y -4 x - 6 y,{x,0,1.5},{y,0,6},RegionFunction->Function[{x,y,z},x +y≤5 && 2x ^2-y≤0&& x≥ 0&& y≥ 0]]

FindMinimum[{2x^2+2y^2-2x y -4 x - 6 y,x +y≤5 && 2x ^2-y≤0 && x≥ 0&& y≥ 0},{{x,0},{y,0.75}}]

FindMinimum[{2x^2+2y^2-2x y -4 x - 6 y,x +y≤5 && 2x ^2-y≤0&& x≥ 0&& y≥ 0},{{x,0},{y,0.75}},StepMonitor:>Print["Step = ","[",x, ",",y,"]"],Method->"InteriorPoint"]

Clear[pts];pts =Reap[FindMinimum[{2x^2+2y^2-2x y -4 x - 6 y,x +y≤5 && 2x ^2-y≤0&& x≥ 0&& y≥ 0},{{x,0},{y,0.75}},StepMonitor:>Sow[{x,y}]]][[2,1]];
pts =Join[{{0,0.75}},pts];
ContourPlot[2x^2+2y^2-2x y -4 x - 6 y,{x,0,4},{y,0,6},Epilog->{Red,Line[pts],Point[pts]},RegionFunction->Function[{x,y,z},x +y≤5 && 2x ^2-y≤0&& x≥ 0&& y≥ 0]]

Clear[pts];pts =Reap[FindMinimum[{2x^2+2y^2-2x y -4 x - 6 y,x +y≤5 && 2x ^2-y≤0&& x≥ 0&& y≥ 0},{{x,0},{y,0}},StepMonitor:>Sow[{x,y}]]][[2,1]];
pts =Join[{{0,0}},pts];
ContourPlot[2x^2+2y^2-2x y -4 x - 6 y,{x,0,4},{y,0,6},Epilog->{Red,Line[pts],Point[pts]},RegionFunction->Function[{x,y,z},x +y≤5 && 2x ^2-y≤0&& x≥ 0&& y≥ 0]]