MAPLE 6
> f:=x^2+y^2+z^2;
2 2 2
f := x + y + z
> g:=x^2+y^2+z^2+x*y*z-10;
2 2 2
g := x + y + z + x y z - 10
> with(linalg):
> Gradf:=grad(f,[x,y,z]);
Gradf := [2 x, 2 y, 2 z]
> Gradg:=grad(g,[x,y,z]);
Gradg := [2 x + y z, 2 y + x z, 2 z + x y]
> SOL:=solve({g=1,seq(Gradf[i]=lambda*Gradg[i],i=1..3)},{x,y,z,lambda});
2
SOL := {x = 0, lambda = 1, z = 0, y = RootOf(-11 + _Z )}, {
3
lambda = 2 RootOf(3 _Z + 7 _Z - 1, label = _L1),
3 2
y = 1 + 7 RootOf(3 _Z + 7 _Z - 1, label = _L1) ,
3 2
x = 1 + 7 RootOf(3 _Z + 7 _Z - 1, label = _L1) ,
3 2
z = 1 + 7 RootOf(3 _Z + 7 _Z - 1, label = _L1) }, {
3
lambda = 2 RootOf(3 _Z + 7 _Z - 1, label = _L1),
3 2
x = 1 + 7 RootOf(3 _Z + 7 _Z - 1, label = _L1) ,
3 2
y = -1 - 7 RootOf(3 _Z + 7 _Z - 1, label = _L1) ,
3 2
z = -1 - 7 RootOf(3 _Z + 7 _Z - 1, label = _L1) }, {
3
lambda = 2 RootOf(3 _Z + 7 _Z - 1, label = _L1),
3 2
y = 1 + 7 RootOf(3 _Z + 7 _Z - 1, label = _L1) ,
3 2
x = -1 - 7 RootOf(3 _Z + 7 _Z - 1, label = _L1) ,
3 2
z = -1 - 7 RootOf(3 _Z + 7 _Z - 1, label = _L1) }, {
3
lambda = 2 RootOf(3 _Z + 7 _Z - 1, label = _L1),
3 2
z = 1 + 7 RootOf(3 _Z + 7 _Z - 1, label = _L1) ,
3 2
x = -1 - 7 RootOf(3 _Z + 7 _Z - 1, label = _L1) ,
3 2
y = -1 - 7 RootOf(3 _Z + 7 _Z - 1, label = _L1) },
2
{y = 0, x = RootOf(-11 + _Z ), lambda = 1, z = 0},
2
{y = 0, z = RootOf(-11 + _Z ), x = 0, lambda = 1}
> allvalues(RootOf(3*_Z+7*_Z^7-1));
7
RootOf(3 _Z + 7 _Z - 1, index = 1),
7
RootOf(3 _Z + 7 _Z - 1, index = 2),
7
RootOf(3 _Z + 7 _Z - 1, index = 3),
7
RootOf(3 _Z + 7 _Z - 1, index = 4),
7
RootOf(3 _Z + 7 _Z - 1, index = 5),
7
RootOf(3 _Z + 7 _Z - 1, index = 6),
7
RootOf(3 _Z + 7 _Z - 1, index = 7)
> allvalues(SOL[6])[1];
{y = 0, lambda = 1, x = sqrt(11), z = 0}
> subs(allvalues(SOL[6]),f);
11
> subs(allvalues(SOL[6]),g);
1