> [!NOTE] Theorem (Nature of Critical Point) > Suppose $f(x,y)$ has a critical point $(a,b)$. Let $D=\det H$ where $H$ is the [[Hessian Matrix|Hessian]] of $f$ given by $H= \begin{pmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{pmatrix},\;D=f_{xx}f_{yy}-f_{xy}^{2}$then > 1. If $D>0$ and $f_{xx}(a,b)>0$, then $(a,b)$ is a [[Local Minimum of Real-Valued Function on Real 2-Space|local minimum]] point. > 2. If $D>0$ and $f_{xx}<0$ at $(a,b)$ then $(a,b)$ is a [[Local Maximum of Real-Valued Function on Real 2-Space|local maximum]] point. > 3. If $D<0$ at $(a,b)$, then $(a,b)$ is a [[Saddle Point of Real-Valued Function on Real 2-Space|saddle]] point. > 4. If $D=0$ then the test is inconclusive. **Proof**. ...