Fermat's theorem (stationary points)FermatsTheoremStationaryPoints2003-07-15 08:36:462008-06-05 06:13:59Theorem% this is the default PlanetMath preamble. as your knowledge
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% define commands hereLet $f\colon (a,b)\to \mathbb R$ be a continuous function and suppose that
$x_0\in (a,b)$ is a local extremum of $f$. If $f$ is differentiable in $x_0$ then $f'(x_0)=0$.
Moreover if $f$ has a local maximum at $a$ and $f$ is differentiable at $a$ (the right derivative exists)
then $f'(a)\le 0$; if $f$ has a local minimum at $a$ then $f'(a)\ge 0$.
If $f$ is differentiable in $b$ and
has a local maximum at $b$ then $f'(b)\ge 0$ while if it has a local minimum at $b$ then $f'(b)\le 0$.