15 - Critical Points

Local Maximum

Let $f(x,y)$ be a function defined on a subset $A\subset {\mathbb{R}}^2$

Point $(x_0,y_0)\in A$ is local maximum of $f$ if there is

$$\text{open disc }{D}_r(x_0,y_0)\text{ of radius }r>0\text{ s.t. }$$ $$f(x,y)\le f(x_0,y_0)\forall (x,y)\in D_r(x_0,y_0)\cap A$$

Note that global maximum is for all $(x,y)\in A$

Local Minimum

Let $f(x,y)$ be a function defined on a subset $A\subset {\mathbb{R}}^2$

Point $(x_0,y_0)\in A$ is local minimum of $f$ if there is

$$\text{open disc }{D}_r(x_0,y_0)\text{ of radius }r>0\text{ s.t. }$$ $$f(x,y)\ge f(x_0,y_0)\forall (x,y)\in D_r(x_0,y_0)\cap A$$

Note that global minimumis for all $(x,y)\in A$