Clear[u1, u2, u3, eq1, eq2, g1, g2, lag1, lag2, lag3, res1, res2, 
      res3, f1, f2, f3];

u1[x_, y_] := y  -  8 (x  -  1/2)^2;

u2[x_, y_] := y  -  8  (x  -  3/2)^2

u3[x_, y_] :=  y  -  (x - 1)^2

g1[x_, y_] :=  4  -  x  - 4 y;

g2[x_, y_] :=  4  -  4 x  -  y;

eq1 = g1[x, y] == 0;

eq2  =  g2[x, y] == 0;

lag1[x1_, x2_, l1_, l2_] :=  
      u1[x1, x2]  +  l1 * g1[x1, x2]  +  l2 * g2[x1, x2];

lag2[x1_, x2_, l1_, l2_] :=  
      u2[x1, x2]  +  l1 * g1[x1, x2]  +  l2 * g2[x1, x2];

lag3[x1_, x2_, l1_, l2_] :=  
      u3[x1, x2]  +  l1 * g1[x1, x2]  +  l2 * g2[x1, x2];

res1 = 
      Solve[D[lag1[x1, x2, l1, l2], #1]&/@{x1, x2, l1, l2} == {0, 0, 0, 
            0}, {x1, x2, l1, l2}]

D[u1[x1, x2], #1]&/@{x1, x2} /. res1

res1 = 
      Solve[D[lag1[x1, x2, l1, l2] /. l2 -> 0, #1]&/@{x1, x2, l1} == {0, 
            0, 0}, {x1, x2, l1}]

D[u1[x1, x2], #1]&/@{x1, x2} /. res1

g1[x1, x2] /. res1

g2[x1, x2] /. res1

(x2 /. Solve[(u1[x1, x2] /. res1[[1]]) == u1[x1, x2], x2] // 
          Simplify)[[1]]

f1[x1_] := (1473/512) - 8 x1 + 8 (x1^2)

graph1 = Plot[f1[x1], {x1, 0, 1}];

graph2 = Plot[-1/4 x1 +  1, {x1, 0, 1}]

graph3 = Plot[4  -  4 x1, {x1, 0, 1}]

Show[graph1, graph2, graph3, AspectRatio -> Automatic]

res2 = 
      Solve[D[lag2[x1, x2, l1, l2], #1]&/@{x1, x2, l1, l2} == {0, 0, 0, 
            0}, {x1, x2, l1, l2}]

D[u2[x1, x2], #1]&/@{x1, x2} /. res2

res2 = 
      Solve[D[lag2[x1, x2, l1, l2] /. l1 -> 0, #1]&/@{x1, x2, l2} == {0, 
            0, 0}, {x1, x2, l2}]

D[u2[x1, x2], #1]&/@{x1, x2} /. res2

g1[x1, x2] /. res2

g2[x1, x2] /. res2

(x2 /. Solve[(u2[x1, x2] /. res2[[1]]) == u2[x1, x2], x2] // 
          Simplify)[[1]]

f2[x1_] := (33/2) - 24 x1 + 8 (x1^2)

graph4 = Plot[f2[x1], {x1, 0, 3/2}]

graph5 = Plot[-1/4 x1 +  1, {x1, 0, 3/2}]

graph6 = Plot[4  -  4 x1, {x1, 0, 3/2}]

Show[graph4, graph5, graph6, AspectRatio -> Automatic]

res3 = 
      Solve[D[lag3[x1, x2, l1, l2], #1]&/@{x1, x2, l1, l2} == {0, 0, 0, 
            0}, {x1, x2, l1, l2}]

D[u3[x1, x2], #1]&/@{x1, x2} /. res3

g1[x1, x2] /. res3

g2[x1, x2] /. res3

(x2 /. Solve[(u3[x1, x2] /. res3[[1]]) == u3[x1, x2], x2] // 
          Simplify)[[1]]

f3[x1_] := (44/25) - 2 x1 + (x1^2)

graph7 = Plot[f3[x1], {x1, 0, 1}]

graph8 = Plot[-1/4 x1 +  1, {x1, 0, 1}]

graph9 = Plot[4  -  4 x1, {x1, 0, 1}]

Show[graph7, graph8, graph9, AspectRatio -> Automatic]