Local Extrema of Multivariable Functions

Definition §

1. $f(x,y)$ has a relative minimum at the point $(a,b)$ if $f(x,y)\ge f(a,b)$ for all points $(x,y)$ in some region around $(a,b)$.

2. $f(x,y)$ has a relative maximum at the point $(a,b)$ if $f(x,y)\le f(a,b)$ for all points $(x,y)$ in some region around $(a,b)$.

Finding §

Local extrema occur at critical points, which satisfy the following condition:

$$\nabla f(a,b)=\vec{0}$$

We must first find the gradient of the function and use the derived system of equations to find the critical points