with(plots):
p1 := polarplot(max(2+cos(x), 2), x = 0 .. 2*Pi, filled = [color=yellow]):
p2 := polarplot(min(2+cos(x), 2), x = 0 .. 2*Pi, filled = [color=white,transparency=0]):
p3 := polarplot([2+cos(x), 2], x = 0 .. 2*Pi, color = [red, blue]):
display( [p3,p2,p1] );

with(plots):
p1 := polarplot(max(2+cos(x), 2), x = 0 .. 2*Pi, filled = [color=yellow]):
p2 := polarplot(min(2+cos(x), 2), x = 0 .. 2*Pi, filled = [color=white,transparency=0]):
p3 := polarplot([2+cos(x), 2], x = 0 .. 2*Pi, color = [red, blue]):
display( [p3,p2,p1] );

... and you must change

if g(a)*g(b) < 0 

to

if evalf(g(a)*g(b)) < 0 

in the loop

... and you must change

if g(a)*g(b) < 0 

to

if evalf(g(a)*g(b)) < 0 

in the loop

@Willem Ottevanger I think that subs is the best possibility because you intend to lose the information that f is a function of x.

@Willem Ottevanger I think that subs is the best possibility because you intend to lose the information that f is a function of x.

Now I understand that it is not possible (at least not efficient) to try to extract points on the surface from the plot structure. Draghilev method (one-man) is a good suggestion.

Now I understand that it is not possible (at least not efficient) to try to extract points on the surface from the plot structure. Draghilev method (one-man) is a good suggestion.

@Carl Love Something got lost after an < -sign!

Implicitplot3d gives a sort of triangularization. You can extract points as follows:

q := plots:-implicitplot3d( x^2+y^3+z^4=1 ,x=-1..1,y=-1..1,z=-1..1 ):
pdata := plottools:-getdata(q); # doesn't work
Q := op(op(1,q));
convert(Q,listlist): R :=(ListTools:-FlattenOnce@@2)(%):
R[129]; #contains [x,y,z,f(x,y,z)]
B := select(A->abs(A[4])<1e-2,R); # only points with f(x,y,z)=0
B[5,1]^2+B[5,2]^3+B[5,3]^4; #check

Your equation is quite unclear. Perhaps you mean something like

diff(z(t),t,t) = (sqrt(diff(z(t)^2,t) + 10^10*z(t)^2)*diff(z(t),t) + 10^10*z(t))*diff(z(t),t);

In this case: rewrite the equation into a system of two equations: diff z(t),t) = y(t), diff(y(t),t) = ...
See ?DEplot

I encountered a similar problem in Ladder animation. Sometimes the whole curve disappears when complex values are calculated.
In your example

plot(Re(sqrt(75-3*x^2)));

will work.

I think it is a bug in Maple 16.

@dipamilo For finding the indices, you must use a simple procedure.
I generalize your problem a little (n rows and m columns in a):

restart; with(ArrayTools): 
n := 2: m := 4:
f:=5*RandomArray(1,n*m);
F := convert(f,Vector):
a:=map(ceil,n*m*RandomArray(n,m));
c_val := seq( min( seq( F[a[j,i]], j=1..n ) ), i=1..m );

indmin := proc(v)
  local k,j;
  k := 1:
  for j from 2 to n do if F[v[j]]<F[v[k]] then k := j end if end do:
  v[k]
end proc:
c_ind := seq( indmin(a[..,i]), i=1..m );

@dipamilo For finding the indices, you must use a simple procedure.
I generalize your problem a little (n rows and m columns in a):

restart; with(ArrayTools): 
n := 2: m := 4:
f:=5*RandomArray(1,n*m);
F := convert(f,Vector):
a:=map(ceil,n*m*RandomArray(n,m));
c_val := seq( min( seq( F[a[j,i]], j=1..n ) ), i=1..m );

indmin := proc(v)
  local k,j;
  k := 1:
  for j from 2 to n do if F[v[j]]<F[v[k]] then k := j end if end do:
  v[k]
end proc:
c_ind := seq( indmin(a[..,i]), i=1..m );

You will see that the circular curve disappears in the last frame. This has to do with rounding errors. The last sqrt is a complex number with a small imaginary part. You can avoid this by replacing the third command by

C := animatecurve( [1/2*p,1/2*Re(sqrt(4-p^2)), p = 0..2], frames=50 ):