extreme values

**alex83** · October 6th 2009, 11:52 AM

hi there

in the shown example (attached ) picture i dont understand the circled phrases in red

what is fxy ? what did we derive to get 1
why is it local maximum??
what is the saddle point in this example

i appreciate your help

**Matt Westwood** · October 6th 2009, 12:22 PM

$f_{xy}$ is $\frac {\partial^2 f}{\partial x \partial y}$ .

That is, you differentiate f partially with respect to x, and then differentiate that partially with respect to y.

When you plug x = -2, y = -2 into the given equation for f, you get 8.

**alex83** · October 6th 2009, 07:17 PM

Originally Posted by Matt Westwood

$f_{xy}$ is $\frac {\partial^2 f}{\partial x \partial y}$ .

That is, you differentiate f partially with respect to x, and then differentiate that partially with respect to y.

thank u i appreciate ur help but i still didnt get it how to derive with respect to (x and y) at the same time.
i got all the other parts (with respect to x then with respect to y).
if i consider both x and y constants i should get f(xy)= 4
is it possible to show me how did we get (1) in that part
thanks

**Matt Westwood** · October 6th 2009, 09:30 PM

Originally Posted by alex83

thank u i appreciate ur help but i still didnt get it how to derive with respect to (x and y) at the same time.
i got all the other parts (with respect to x then with respect to y).
if i consider both x and y constants i should get f(xy)= 4
is it possible to show me how did we get (1) in that part
thanks

You have your equation (which I can't see in front of me because you didn't type it in just copied the page in and I can't see it when I'm editing it).

Partially differentiate it with respect to $x$ (partially) to get the expression for $f_x$ . That is, $f_x = \frac {\partial f}{\partial x}$ . It's just a more compact notation for it.

Then differentiate that partially with respect to $y$ to get $f_{xy}$ .

That is, $f_{xy}$ is $(f_x)_y$ or $\frac {\partial}{\partiial y} \left({\frac {\partial f}{\partial x}}\right)$ .

And when you work it out (they've done $f_x$ for you) you get $f_{xy} = 1$ .

You'll probably find something about this in your text book.

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