Fermat's stationary point theorem states that if a real-valued function 
of a real variable has a
local extremum at a point 
in the interior of its domain
and is differentiable at 
, then
 |
The illustration shows the horizontal tangents at a differentiable local minimum and local
maximum, together with a stationary point
that is an inflection point. The converse
is false: a stationary point need not be a local maximum or local
minimum. For example, 
has 
,
but no local extremum at 0. The hypotheses also
matter. The nondifferentiable function 
has a local minimum
at 0, where 
is the absolute value
of 
.
