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I'm having trouble with the following question about local maxima and minima.

Any help is appreciated. Thanks.

Show that if $a > b > c > 0$ than the function

$$f(x,y,z) = (ax^2 +by^2 +cz^2) e^{-x^2 -y^2 -z^2}$$

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

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I suspect you know how to start. What is the derivative? When it is equal to 0? Where is your work? – mixedmath Mar 16 '12 at 15:52
It's also important to determine if the function is defined on a compact subset of R^3.If so,it has absolute maxima and minima. By the way the problem is worded,I assume not. – Mathemagician1234 Mar 16 '12 at 16:36
Note: "maximum" and "minimum" are singular; "maxima" and "minima" are plural. So "There is one local maximum" or "There are two local maxima" would be correct usage. (I fixed these in the posting.) – Michael Hardy Mar 17 '12 at 1:02
up vote 0 down vote accepted

The maximum and minimum is attend when $df=0$, mybe you can use also the Hessian matrix (you can know if it's a Maxima or Minima by the Hessian matrix)

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Could you please explain this? I dont understand – Euden Mar 18 '12 at 12:55
you find the points X who's $df=0$ If the Hessian is positive definite at x, then f attains a local minimum at x. If the Hessian is negative definite at x, then f attains a local maximum at x – Abdelmajid Khadari Mar 28 '12 at 12:03

At any of these points you need to have $\frac{\partial f}{\partial x} = 0$ and same for the remaining variables.

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@Abdelmajid, Re: your suggested edit - by "same for the remaining variables" this user means $\partial f/ \partial y$ etc. also vanish, not only wrt (with respect to) $x$. – anon Mar 17 '12 at 13:46

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