First and Second Derivative Tests

3 ways to determine what kind of critical value occurs when f'(c) = 0

1) First derivative test
If f' goes from + to -, it is a maximum. If f' goes from - to +, it is a minimum.

2) Test points near c

3)Second derivative test
If f''(c) > 0, it is a minimum
If f''(c) < 0, it is a maximum.

The second partials test:
Let f(x,y) have continuous first and second partial derivatives
If (xo, yo) is a critical point, consider
d = fxx(xo,yo) fyy(xo,yo) - [fxy(xo,yo)^2

If d>0 and fxx>0, then relative minimum
If d>0 and fxx<0, then relative maximum
If d<0, then saddle point
If d=0, inconclusive