4.1 Maxima and minima

$\bullet$ A local maximum is like the top of a hill.

$\bullet$ A local minimum is like the bottom of a bowl.

Let $f(x,y)$ be a differentiable function of two variables. See the moodle page for diagrams. We say that $f$ has a local maximum at $(a,b)$ if

for all $(x,y)$ close to $(a,b)$; likewise we say that $f$ has a local minimum at $(a,b)$ if

for all $(x,y)$ close to $(a,b)$. These points are often of interest, and we need a way of locating them.