- #1

azatkgz

- 186

- 0

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