Lagrange multiplier pictures
Here are some pictures together with the Maple commands which generated them. I typed with(plots): before all of them.

I strongly urge people who are interested to invoke their own copy of Maple and look at the pictures! My own choices of perspective and (ugh!) color will not likely be best for you. Also, you can rotate the pictures in Maple as you wish, and gain more insight. My experience tells me that "intuition" is obtained by working with examples. The labels (Fig.1, etc.) refer to those in the Lagrange multiplier discussion.

A:=implicitplot3d(x^2+y^2+z^2=1, x=-1.5..1.5, y=-1.5..1.5, z=-1.5..1.5, color=brown, axes=normal, scaling=constrained, grid=[20,20,20]): implicitplot3d(x*y*z=0, x=-1.75..1.75, y=-1.75..1.75, z=-1.75..1.75, color=pink, axes=normal, scaling=constrained, grid=[20,20,20]);
Picture #1Picture #2
TURF:=v->implicitplot3d(x*y^2*z^3=v, x=-1.8..1.8, y=-1.8..1.8, z=-1.8..1.8, color=blue, axes=normal, scaling=constrained, grid=[25,25,25]);
TURF(-.05); 		display3d({A,TURF(-sqrt(3)/36)});
Picture #3Picture #4
SURF:=(v,w)->implicitplot3d(x+y^2+z^3=v, x=-2.5..2.5, y=-2.5..2.5, z=-2.5..2.5, color=w, axes=normal, scaling=constrained, grid=[25,25,25]);
Here I made the Maple function depend both on the constant for the level surface, and the color. That's so I could vary the color easily in the next display.
display3d({SURF(6,green), SURF(.5,pink), SURF(-5,yellow)}); 		display3d({A,SURF(-1.17205,pink)});
Picture #5Picture #6
display3d({A,SURF(1.25,pink)});
Picture #7Picture #8
LA:=implicitplot3d(x^2+y^2+z^2=1, x=0..1.5, y=0..1.5, z=0..1.5, color=blue, axes=normal, scaling=constrained, grid=[30,30,30]):
LSURF:=v->implicitplot3d(x+y^2+z^3=v, x=0..2.5, y=0..2.5, z=0..2.5, color=yellow, axes=normal, scaling=constrained, grid=[30,30,30]);
display3d({LA,LSURF(1.102)});
Picture #9