CriticalPointTangentPlane::usage="
References:Belk,Jim.UniversityofGlasgow.Homework9,Prob.-1;Personal.";
Development:
(a)Critical point :
In[]:=
X[u_,v_]:={
2
u
+6v,3u+9v,
2
u
+2u+
2
v
};​​Eliminate[{x==
2
u
+6v,y==3u+9v,z==
2
u
+2u+
2
v
},{u,v}]​​
-55404x+8100
2
x
+36936y-7776xy+2268
2
y
+144x
2
y
-108
3
y
+
4
y
-55404z-14580xz+10692yz-162
2
y
z+6561
2
z
0
In[]:=
Solve[-55404x+8100
2
x
+36936y-7776xy+2268
2
y
+144x
2
y
-108
3
y
+
4
y
-55404z-14580xz+10692yz-162
2
y
z+6561
2
z
0,z]
z
1
81
342+90x-66y+
2
y
-2
3
9747+9747x-6840y-342xy+231
2
y
+3x
2
y
-2
3
y
,z
1
81
342+90x-66y+
2
y
+2
3
9747+9747x-6840y-342xy+231
2
y
+3x
2
y
-2
3
y
;
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 critical points"]];​​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
1
81
342+90x-66y+
2
y
-2
3
9747+9747x-6840y-342xy+231
2
y
+3x
2
y
-2
3
y
,{-15,15},{-15,15}
Minimum: {1,-3,-1}
Out[]=
Null
(b) Tangent plane at the minimum point::
In[]:=
PlaneFunction[func_,{x0_,y0_,z0_},{u0_,u1_},{v0_,v1_}]:=​​Module[{},​​g1=func/.{xx0,yy0};​​If[N[u0]>=N[u1]||N[v0]>=N[v1]||N[z0]=!=N[g1],​​Print["Incorrect intervals or the point does not belong to the function"],​​a1=Limit[((func/.{xx,yy0})-z0)/(x-x0),xx0];​​b1=Limit[((func/.{xx0,yy})-z0)/(y-y0),yy0];​​c1=-1;​​d1={a1,b1,c1}.{x0,y0,z0};​​​​Print["The equation of the tangent plane is: ",{x,y,z}.{a1,b1,c1}," = ",d1];​​Plot3D[{func,d1/c1-(a1/c1)x-(b1/c1)y},{x,u0,u1},{y,v0,v1}]]​​​​]
In[]:=
PlaneFunction
1
81
342+90x-66y+
2
y
-2
3
9747+9747x-6840y-342xy+231
2
y
+3x
2
y
-2
3
y
,{1,-3,-1},{-15,15},{-15,20}
The equation of the tangent plane is: -z = 1
Out[]=