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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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    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.)2012-03-17

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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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    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 x2012-03-28
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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) $x$.2012-03-17