10 - Type of Critical Point from Partial Derivatives

Type of Critical Point from Partial Derivatives

Suppose that $f(x,y)$ is defined on an open subset $U$ with

• continuous partial derivatives

• $(x_0,y_0)\in U$ is a critical point ($\nabla f(x_0,y_0)=0$)

1. If

$$\frac{{\partial }^{2}f}{\partial x^2}(x_0,y_0)\frac{{\partial }^{2}f}{\partial y^2}(x_0,y_0)-{\left(\frac{{\partial }^{2}f}{\partial x\partial y}(x_0,y_0)\right)}^2>0,\frac{{\partial }^{2}f}{\partial x^2}(x_0,y_0)<0$$

Then $(x_0,y_0)$ is a local maximum

2. If

$$\frac{{\partial }^{2}f}{\partial x^2}(x_0,y_0)\frac{{\partial }^{2}f}{\partial y^2}(x_0,y_0)-{\left(\frac{{\partial }^{2}f}{\partial x\partial y}(x_0,y_0)\right)}^2>0,\frac{{\partial }^{2}f}{\partial x^2}(x_0,y_0)>0$$

Then $(x_0,y_0)$ is a local maximum

Proof

Classifying Stationary Points

All stationary points have $f_x=0$ and $f_y=0$ with

1. If $f_{xx}{f}_{yy}-f_{xy}^2>0$ and $f_{xx}>0$ (or $f_{yy}>0$) then

$$\text{local maximum}$$

2. If $f_{xx}{f}_{yy}-f_{xy}^2>0$ and $f_{xx}<0$ (or $f_{yy}<0$) then

$$\text{local maximum}$$

3. If $f_{xx}{f}_{yy}-f_{xy}^2<00$ then

$$\text{saddle point}$$