- #1

xokaitt

- 6

- 0

## Homework Statement

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!