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Local maxima, minima, and saddle points

Show that if a>b>c>0 then the function:

f(x,y,z)=(ax2+by2+cz2)exp⁡(−x2−y2−z2) f(x,y,z) = (ax^2 +by^2 + cz^2 )\exp(-x^2-y^2-z^2)

Can someone please help me with this question I have spent hours on it and have made no progress..
Reply 1
Avatar for gff
gff
5
Then the function what? :tongue:
Reply 2
Avatar for Neilding91
Neilding91
OP
Show that if a>b>c>0 then the function:

f(x,y,z)=(ax2+by2+cz2)exp⁡(−x2−y2−z2) f(x,y,z) = (ax^2 +by^2 + cz^2 )\exp(-x^2-y^2-z^2)

has two local maxima, one local minima and four saddle points.

Can someone please help me with this question I have spent hours on it and have made no progress..
Reply 3
Avatar for gff
gff
5
I reckon there is more to it than the second derivative test and algebra. However, I can't think of the right idea yet. :tongue:
Reply 4
Avatar for Slowbro93
Slowbro93
20
What working out have you done so far?
Reply 5
Avatar for ztibor
ztibor
10
Original post by Neilding91
Show that if a>b>c>0 then the function:

f(x,y,z)=(ax2+by2+cz2)exp⁡(−x2−y2−z2) f(x,y,z) = (ax^2 +by^2 + cz^2 )\exp(-x^2-y^2-z^2)

Can someone please help me with this question I have spent hours on it and have made no progress..


1. write down the partal derivatives and take them equal zero.
2. Solve them simoultaneously
a) first solution is trivial (0,0,0)
b)then arrange the first equation to x^2 and substitute into the others
You will get y=0 and z=0 so x^2=1
this gives 2 soluutions
c) because of simmetry similary to above for y and z you will get more 4 solutions
So you have 7 stationary points
3. Determine the second partial derivatives
4. Create the Hessian matrix for each given (x,y,z) value (7 matrices)
5. For every matrix:
Calculate the eigen values
a) When the matrix positive definite (all eigen value is positive)
then there minimum at that point where the matrix was calculated
b) when the matrix negative definite there is maximum
c) and when the matrix has both negative and positive eigen values or zero
then there is a saddle at the given point
(edited 12 years ago)
Reply 6
Avatar for gff
gff
5
Original post by ztibor
...


That looks like my shopping list -- can't follow it either; too long. :biggrin:
Reply 7
Avatar for Fat-Love
Fat-Love
15
I haven't tried it yet but I cant see why brute force wouldn't work in trying to calculate the maxima and minima etc.

EDIT: as ztbor did. Sounds like a reasonable method.
(edited 12 years ago)
Reply 8
Avatar for ztibor
ztibor
10
Original post by gff
That looks like my shopping list -- can't follow it either; too long. :biggrin:


... can you suggest what to follow
According the theorem for existence of stationary point in multivariable functions:
grad(f(r⃗))=0⃗grad( f(\vec r))=\vec 0 (depending on that grad(f) is exists)
To determine the type of this point you have to use the eigen values for Hessian matrix.
(edited 12 years ago)

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