Which conditions guarantee a local maximum at a critical point?

• A. $f_{xx} < 0$
• C. $D = f_{xx} f_{yy} - f_{xy}^2 > 0$
• D. $f_x = 0$ and $f_y = 0$ (first-order conditions)

All three are needed: zero gradient (it's a critical point), positive determinant (not a saddle), and negative $f_{xx}$ (concave, so it's a max, not a min).