Calculating critical points and classifying them

[Gekko]

Homework Statement

Find all critical points and classify them

Homework Equations

f(x,y) = sin(x)sin(y)sin(x+y)

0<=x,y<=Pi

The Attempt at a Solution

fx=sinysin(2x+y) and fy=sinxsin(2y+x)

Therefore critical points are at:

x=Pi/3(2n-m) , y=Pi/3(2m-n) where n>=1, m<=2, n,m belong to integer set (Z)

fxx = sin(2x+2y)-sin(2x)
fyy = sin(2x+2y)-sin(2y)
fxy = sin(2x+2y)

Now, to classify the critical points I was simply going to test for:

fxxfyy-fxy^2

If <0 it is a saddle point etc and substitute for x the definitions above to obtain the equation we can use to classify with

Is this correct? Is there a better way? Would greatly appreciate comments. Thanks

 

[Raskolnikov]

Yep, that's the second derivative test. So it's definitely correct (assuming you have that "etc" part right in your comment). As for a better way...there might be something intuitive about this function in particular that makes it special. I don't see anything, but I'm not sure. However, in general, the second derivative test is the way to go.