PROCEDURE:

- Calculate $f_x, f_y, f_{xx}, f_{xy}, f_{yy}$

- Calculate the critical points ;

Then for each critical point:

- Calculate $A = f_{xx}(x_0, y_0)$

- Calculate $B = f_{xy}(x_0, y_0)$

- Calculate $C = f_{yy}(x_0, y_0)$

- $AC-B^2$.

Apply the Second Derivative Test

If $AC-B^2 > 0$ and $A > 0$ minimum

If $AC-B^2 > 0$ and $A < 0$ maximum

If $AC-B^2 < 0$ saddle

If $AC-B^2 = 0$ degenerate

Click to enlarge

EXAMPLE:

( See Mathematica print )

$f(x,y)=e^{x^2-\frac{x^4}{4}-y^2}$

At $(0,0): \left\{AC-B^2, A\right\} = \left\{-4,2\right\}$

Saddle.

At $(-\sqrt{2},0): \left\{AC-B^2, A\right\} = \left\{8e^2,-4e\right\}$

Local maximum.

At $(\sqrt{2},0): \left\{AC-B^2, A\right\} = \left\{8e^2,-4e\right\}$

Local maximum.

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