ParabolaGeneratesSurface::usage="
​Referencia: Belk, Jim. University of Glasgow. Homework 9, Prob.- 2";
(a) Parameterization of the

X
surface:
In[]:=
Manipulate[Show[{RevolutionPlot3D[{
2
z
-1,0,z},{z,-3,3},{θ,0,θ1},RevolutionAxis{0,0,1},PlotStyle->Opacity[0.6],MeshFalse,BoundaryStyleDirective[Black,Thick],PerformanceGoal"Quality",ColorFunction"BlueGreenYellow"],ParametricPlot3D[t{0,0,1},{t,-5,5},ColorFunction->"BlueGreenYellow"],Graphics3D[{Black,Ball[{0,0,0},0.15]}],Graphics3D[{Black,Ball[{Cos[θ1-π],Sin[θ1-π],0},0.15]}],​​Graphics3D[{Black,Dashed,Thick,Line[{{0,0,0},{Cos[θ1-π],Sin[θ1-π],0}}]}]},AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange8,ViewPoint{1.3,-1,0.5},ImageSize900],Style["Surface of revolution:",Bold,Medium],{{θ1,0.00002,"Value (θ1)"},0.00001,2π,0.00001},ControlPlacementTop]
Out[]=
​
Surface of revolution:
Value (θ1)
Cartesian equation of surface
±
2
x
+
2
y
==
2
z
-1;​​
2
x
+
2
y
==
4
z
-2
2
z
+1;​​
2
x
+
2
y
-
4
z
+2
2
z
-1==0;
The parameterization results:

X
[u_,v_]:=u,v,±
1±
2
u
+
2
v
;​​
When the sign that precedes the radical
2
u
+
2
v
it is + , the radical is positive, u and v take any real value.When the sign that precedes the radical
2
u
+
2
v
it is - , the term
2
u
+
2
v
it will have to be
2
u
+
2
v
≤ 1.
(b) Critical points of

X
and what happens at these points?:
In[]:=
Minimize
1+
2
x
+
2
y
,{x,y}
Out[]=
{1,{x0,y0}}
In[]:=
Plot3D
1+
2
x
+
2
y
,{x,-3,3},{y,-3,3}
Out[]=
Minimum:
{0,0,1}
In[]:=
Maximize
1-
2
x
+
2
y
,{x,y}
Out[]=
{1,{x0,y0}}
In[]:=
Plot3D
1-
2
x
+
2
y
,{x,-3,3},{y,-3,3}
Out[]=
Maximum:
{0,0,1}
In[]:=
Minimize-
1-
2
x
+
2
y
,{x,y}
Out[]=
{-1,{x0,y0}}
In[]:=
Plot3D-
1-
2
x
+
2
y
,{x,-3,3},{y,-3,3}
Out[]=
Minimum:
Maximum:
Conclusion:
Each function must be analyzed separately.