Study guide and 2 practice problems on:

Second derivative test for functions of several variables

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Consider a critical point of the differentiable function $f(x,y)$.
Let $D = \text{det} \begin{pmatrix} f_{x x} & f_{x y}\\ f_{y x} & f_{y y} \end{pmatrix} = f_{x x} \cdot f_{y y} - f_{x y}^2 $.
If $D > 0$ and $f_{x x} > 0$ at a critical point, then it is a minimum.
If $D >0$ and $f_{x x} < 0$ at a critical point, then it is a maximum.
If $D < 0$, at a critical point, it is a saddle point.

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