(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let f(x,y)=Ax^{2}+E where A and E are constants. What are the critical points of f(x,y)? Determine whether the critical points are local maxima, local minema, or saddle points.

2. The attempt at a solution

First I found the first partial derivatives with respect to x and y

[tex]\partial[/tex]f/[tex]\partial[/tex]x=2Ax

[tex]\partial[/tex]f/[tex]\partial[/tex]y=0

[tex]\Rightarrow[/tex] 2Ax=0,

[tex]\Rightarrow[/tex] x=0 for any constant A.

Therefore, all points lying on the y-axis are critical points.

(i.e. C.P.'s = (0,n), n[tex]\in[/tex]R.)

Now, we have to find the second partial's with respect to x and y.

[tex]\partial[/tex]^{2}f/[tex]\partial[/tex]x^{2}=2A

[tex]\partial[/tex]^{2}f/[tex]\partial[/tex]y^{2}=0

and

[tex]\partial[/tex]^{2}f/[tex]\partial[/tex]x[tex]\partial[/tex]y=0

Therefore Df=([tex]\partial[/tex]^{2}f/[tex]\partial[/tex]x^{2})([tex]\partial[/tex]^{2}f/[tex]\partial[/tex]y^{2})-([tex]\partial[/tex]^{2}f/[tex]\partial[/tex]x[tex]\partial[/tex]y)^{2}at (0,n) , n[tex]\in[/tex]R.

[tex]\Rightarrow[/tex] Df=(2A)(0)-(0)^{2}=0

This is where I get stuck. Now that Df=0, how do I determine whether or not the critical pts are local extrema or saddle pts?

From plotting the function on Mathematica, I know that these critical points are in fact saddle points, but I don't know how to mathematically state that.

Thanks!

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# Homework Help: Finding the critical points of a multivariable function and determining local extrema

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