Find Critical Points of Hessian Matrix

[azatkgz]

Please,check my solution.
Find critical points of the function f(x,y,z)=x^3+y^2+z^2+12xy+2z
and determine their types (degenerate or non-degenerate, Morse index for non-
degenerate).

Attempt

\frac{df}{dx}=3x^2+12y=0

\frac{df}{dy}=2y+12x=0

\frac{df}{dz}=2z+2=0

Critical points are at

x=24 y=-144 z=-1

x=0 y=0 z=-1

H(f)=\left|\begin{array}{l[cr]}6x&12&0\\12&2&0\\0&0&2\end{array}\right|


for x=24

det\left|\begin{array}{l[cr]}144&12&0\\12&2&0\\0&0&2\end{array}\right|=288 non-degenerate


for x=0


det\left|\begin{array}{l[cr]}0&12&0\\12&2&0\\0&0&2\end{array}\right|=-288 non-degenerate

 

[Marco_84]

Please,check my solution.
Find critical points of the function f(x,y,z)=x^3+y^2+z^2+12xy+2z
and determine their types (degenerate or non-degenerate, Morse index for non-
degenerate).

Attempt

\frac{df}{dx}=3x^2+12y=0

\frac{df}{dy}=2y+12x=0

\frac{df}{dz}=2z+2=0

Critical points are at

x=24 y=-144 z=-1

x=0 y=0 z=-1

H(f)=\left|\begin{array}{l[cr]}6x&12&0\\12&2&0\\0&0&2\end{array}\right|


for x=24

det\left|\begin{array}{l[cr]}144&12&0\\12&2&0\\0&0&2\end{array}\right|=288 non-degenerate


for x=0


det\left|\begin{array}{l[cr]}0&12&0\\12&2&0\\0&0&2\end{array}\right|=-288 non-degenerate

keep going... are they minimum, maximum.. saddle points??

 

[azatkgz]

Morse index

for (0,0,-1)

det\left|\begin{array}{l[cr]}-\lambda &12&0\\12&2-\lambda &0\\0&0&2-\lambda\end{array}}\right|=0

(2-\lambda )(-\lambda(2-\lambda)-144)=0

\lambda_1=2,\lambda_2=1-\sqrt{145},\lambda_3=1+\sqrt{145}


for (24,-144,-1)



det\left|\begin{array}{l[cr]}144-\lambda &12&0\\12&2-\lambda &0\\0&0&2-\lambda\end{array}}\right|=0

(2-\lambda )((144-\lambda)(2-\lambda)-144)=0

\lambda_1=2,\lambda_2=73-\sqrt{5185},\lambda_3=73+\sqrt{5185}

Is it right?What we can say about maximum,minimum and saddle points?

 

[Marco_84]

Morse index

for (0,0,-1)

det\left|\begin{array}{l[cr]}-\lambda &12&0\\12&2-\lambda &0\\0&0&2-\lambda\end{array}}\right|=0

(2-\lambda )(-\lambda(2-\lambda)-144)=0

\lambda_1=2,\lambda_2=1-\sqrt{145},\lambda_3=1+\sqrt{145}


for (24,-144,-1)



det\left|\begin{array}{l[cr]}144-\lambda &12&0\\12&2-\lambda &0\\0&0&2-\lambda\end{array}}\right|=0

(2-\lambda )((144-\lambda)(2-\lambda)-144)=0

\lambda_1=2,\lambda_2=73-\sqrt{5185},\lambda_3=73+\sqrt{5185}

Is it right?What we can say about maximum,minimum and saddle points?

i didnt check you're calculus, but find what the sign of eigenvalues mean and you'll get you're answer.

bye