CriticalPointsSurfaceCurvatures::usage="
​References: Belk, Jim. University of Glasgow. Homework 12, Prob.- 2.";
Development:
(a) Critical points:
In[]:=
Maxymin[func_,{x0_,x1_},{y0_,y1_}]:=​​Module[{},​​If[N[x0]>=N[x1]||N[y0]>=N[y1],Print["Incorrect intervals"],​​f[x_,y_]:=func;​​e1=D[f[x,y],x];​​e2=D[f[x,y],y];​​e11=D[e1,x];​​e12=D[e1,y];​​e22=D[e2,y];​​solución=Solve[{e1==0,e2==0},{x,y}];​​​​x11=x/.solución;y11=y/.solución;​​n=Length[x11];​​Do[x2=Take[x11,{j}];​​y2=Take[y11,{j}];​​xc=Dot[x2,{1}];yc=Dot[y2,{1}];​​If[N[Abs[Im[xc]]+Abs[Im[yc]]]==0,​​g=e11/.{xxc,yyc};​​h=e12/.{xxc,yyc};​​k=e22/.{xxc,yyc};​​i=g*k-h^2;​​Print[""];​​If[N[i]>0&&N[g]<0,Print["Maximum: ",{xc,yc,func/.{xxc,yyc}}]];​​If[N[i]>0&&N[g]>0,Print["Minimum: ",{xc,yc,func/.{xxc,yyc}}]];​​If[N[i]==0,Print["It has no extreme value or saddle point"]];​​If[N[i]<0,Print["Saddle point: ",{xc,yc,func/.{xxc,yyc}}]],​​Print["................."]],{j,1,n}]]​​Plot3D[f[x,y],{x,x0,x1},{y,y0,y1}]]​​
In[]:=
Maxymin[3
2
x
+4Cos[x+y],{-5,5},{-5,5}]
Saddle point: {0,0,4}
Out[]=
Null
(b) Curvatures:
In[]:=
ResourceFunction["PrincipalCurvatures"][3
2
x
+4Cos[x+y],{x,y,z}]
In[]:=
Simplify3
2
(3
2
x
Cos[x+y]-2
2
Sin[x+y]
)
3
(9
2
x
-12xSin[x+y]+8
2
Sin[x+y]
)
+((6x-4Sin[x+y])((6-4Cos[x+y])(6x-4Sin[x+y])+16Cos[x+y]Sin[x+y])-4Sin[x+y](-4Cos[x+y](6x-4Sin[x+y])+16Cos[x+y]Sin[x+y])-(6-8Cos[x+y])(
2
(6x-4Sin[x+y])
+16
2
Sin[x+y]
))2
3/2
(
2
(6x-4Sin[x+y])
+16
2
Sin[x+y]
)
,-3
2
(3
2
x
Cos[x+y]-2
2
Sin[x+y]
)
3
(9
2
x
-12xSin[x+y]+8
2
Sin[x+y]
)
+((6x-4Sin[x+y])((6-4Cos[x+y])(6x-4Sin[x+y])+16Cos[x+y]Sin[x+y])-4Sin[x+y](-4Cos[x+y](6x-4Sin[x+y])+16Cos[x+y]Sin[x+y])-(6-8Cos[x+y])(
2
(6x-4Sin[x+y])
+16
2
Sin[x+y]
))2
3/2
(
2
(6x-4Sin[x+y])
+16
2
Sin[x+y]
)

In[]:=
3
2
(3
2
x
Cos[x+y]-2
2
Sin[x+y]
)
+3
2
x
Cos[x+y]-2
2
Sin[x+y]
,-3-3
2
x
Cos[x+y]+2
2
Sin[x+y]
+
2
(3
2
x
Cos[x+y]-2
2
Sin[x+y]
)
3/2
(9
2
x
-12xSin[x+y]+8
2
Sin[x+y]
)
/.{x0,y0,z4}
Out[]=
{0,0}
In[]:=
ResourceFunction["GaussianCurvature"][3
2
x
+4Cos[x+y],{x,y,z}]
Out[]=
0
In[]:=
ResourceFunction["MeanCurvature"][3
2
x
+4Cos[x+y],{x,y,z}]
In[]:=
((6x-4Sin[x+y])((6-4Cos[x+y])(6x-4Sin[x+y])+16Cos[x+y]Sin[x+y])-4Sin[x+y](-4Cos[x+y](6x-4Sin[x+y])+16Cos[x+y]Sin[x+y])-(6-8Cos[x+y])(
2
(6x-4Sin[x+y])
+16
2
Sin[x+y]
))/.{x0,y0,z4}
Out[]=
0