- #1
azatkgz
- 186
- 0
Please,check my solution.
Find critical points of the function [tex]f(x,y,z)=x^3+y^2+z^2+12xy+2z[/tex]
and determine their types (degenerate or non-degenerate, Morse index for non-
degenerate).
Attempt
[tex]\frac{df}{dx}=3x^2+12y=0[/tex]
[tex]\frac{df}{dy}=2y+12x=0[/tex]
[tex]\frac{df}{dz}=2z+2=0[/tex]
Critical points are at
x=24 y=-144 z=-1
x=0 y=0 z=-1
[tex]H(f)=\left|\begin{array}{l[cr]}6x&12&0\\12&2&0\\0&0&2\end{array}\right|[/tex]
for x=24
[tex]det\left|\begin{array}{l[cr]}144&12&0\\12&2&0\\0&0&2\end{array}\right|=288[/tex] non-degenerate
for x=0
[tex]det\left|\begin{array}{l[cr]}0&12&0\\12&2&0\\0&0&2\end{array}\right|=-288[/tex] non-degenerate
Find critical points of the function [tex]f(x,y,z)=x^3+y^2+z^2+12xy+2z[/tex]
and determine their types (degenerate or non-degenerate, Morse index for non-
degenerate).
Attempt
[tex]\frac{df}{dx}=3x^2+12y=0[/tex]
[tex]\frac{df}{dy}=2y+12x=0[/tex]
[tex]\frac{df}{dz}=2z+2=0[/tex]
Critical points are at
x=24 y=-144 z=-1
x=0 y=0 z=-1
[tex]H(f)=\left|\begin{array}{l[cr]}6x&12&0\\12&2&0\\0&0&2\end{array}\right|[/tex]
for x=24
[tex]det\left|\begin{array}{l[cr]}144&12&0\\12&2&0\\0&0&2\end{array}\right|=288[/tex] non-degenerate
for x=0
[tex]det\left|\begin{array}{l[cr]}0&12&0\\12&2&0\\0&0&2\end{array}\right|=-288[/tex] non-degenerate