Min, Max, or Saddle Point at (0,0) for F=?

[tatianaiistb]

Homework Statement

Decide between a minimum, maximum, or saddle point for:

(a) F=-1+4(e^x-x)-5xsin(y)+6y² at the point x=y=0

(b) F=(x²-2x)cos(y) with stationary point at x=1, y=pi

The professor was a bit confusing, so I did it the way I remember from Calc, but am unsure of whether I went the right direction or not... Ideas?

Homework Equations

F_xxF_yy-F_xy²

The Attempt at a Solution

For part a,

I got the point is at a minimum because
F_xxF_yy-F_xy²=23, which is greater than 0, and so are both F_xx and F_yy (4 and 12 respectively).

For part b,

I got the point is at a maximum because
F_xxF_yy-F_xy²=2, which is greater than 0, but both F_xx and F_yy are negative (-2 and -1 respectively).

Am I thinking correctly? Thanks!

 

[I like Serena]

Yes, you're thinking correctly.
For reference, you've been applying the Second partial derivative test:
http://en.wikipedia.org/wiki/Second_partial_derivative_test

Note that you don't have to check the sign of F_yy.

Btw, you should also check that the first derivatives are zero.