> restart:
> Digits:=20:
> G1:=2-x2+exp(-0.1*x1^2)+(0.2*x1)^4:
> G2:=4.5-x1*x2;
> G:=min(G1,G2);
> with(plots):
> p1:=implicitplot(G1,x1=-5..5,x2=-5..5,grid=[101,101]):
> p2:=implicitplot(G2,x1=-5..5,x2=-5..5,grid=[101,101],color=blue):
> p3:=implicitplot(G=0,x1=-5..5,x2=-5..5,grid=[101,101],color=green,thic
> kness=4):
> with(plottools):
> c := circle([0,0], 3, color=magenta):
> display([p1,p2,p3,c]);
> evalf(subs(x1=0,x2=0,G));

                          G2 := 4.5 - x1 x2


                                   2             4
      G := min(2 - x2 + exp(-0.1 x1 ) + 0.0016 x1 , 4.5 - x1 x2)

Warning, the name changecoords has been redefined

Warning, the name arrow has been redefined



                                  3.

> y1:=solve(G1,x2);
> y2:=solve(G2,x2);
> xL:=fsolve(y1=y2,x1);

                                           2
  y1 := 2. + exp(-0.10000000000000000000 x1 )

                                      4
         + 0.0016000000000000000000 x1


                           4.5000000000000000000
                     y2 := ---------------------
                                    x1


                     xL := 1.6183841765382898291

> Pdef:=(Int(Int(1/(2*Pi)*exp(-(x1^2+x2^2)/2),x2=-infinity..y2),x1=-infi
> nity..0))+(Int(Int(1/(2*Pi)*exp(-(x1^2+x2^2)/2),x2=y1..infinity),x1=-i
> nfinity..xL))+(Int(Int(1/(2*Pi)*exp(-(x1^2+x2^2)/2),x2=y2..infinity),x
> 1=xL..infinity));

             0
            /
           |
  Pdef :=  |
           |
           |
          /
            -infinity

           4.5000000000000000000
           ---------------------             2     2
                    x1                     x1    x2
          /                          exp(- --- - ---)
         |                                  2     2
         |                       1/2 ---------------- dx2 dx1 +
         |                                  Pi
         |
        /
          -infinity

           1.6183841765382898291                 2     2
          /                                    x1    x2
         |                               exp(- --- - ---)
         |                                      2     2
         |                       Int(1/2 ----------------, x2 = 2.
         |                                      Pi
         |
        /
          -infinity

                                         2
         + exp(-0.10000000000000000000 x1 )

                                      4
         + 0.0016000000000000000000 x1  .. infinity) dx1 +

           infinity
          /
         |
         |
         |
         |
        /
          1.6183841765382898291

                                            2     2
           infinity                       x1    x2
          /                         exp(- --- - ---)
         |                                 2     2
         |                      1/2 ---------------- dx2 dx1
         |                                 Pi
         |
        /
          4.5000000000000000000
          ---------------------
                   x1

> Pdef_num:=evalf(Pdef);

                 Pdef_num := 0.0034789463209310872097

> PHI:=proc(x)
>   int(1/sqrt(2*Pi)*exp(-t^2/2),t=-infinity..x)
> end:
> -fsolve(PHI(x)=Pdef_num,x);

                        1.2559685128742776753

> 
> 1-7.1529e-3/9.81929e-3;

                        0.27154610974927922487

> Digits:=20:evalf(sqrt(1/2));

                        0.70710678118654752440

> 
