# Fermat’s theorem (stationary points)

Let $f\colon(a,b)\to\mathbb{R}$ 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 $f^{\prime}(x_{0})=0$.

Moreover if $f$ has a local maximum at $a$ and $f$ is differentiable at $a$ (the right derivative exists) then $f^{\prime}(a)\leq 0$; if $f$ has a local minimum at $a$ then $f^{\prime}(a)\geq 0$. If $f$ is differentiable in $b$ and has a local maximum at $b$ then $f^{\prime}(b)\geq 0$ while if it has a local minimum at $b$ then $f^{\prime}(b)\leq 0$.

Title Fermat’s theorem (stationary points) FermatsTheoremstationaryPoints 2013-03-22 13:45:05 2013-03-22 13:45:05 paolini (1187) paolini (1187) 7 paolini (1187) Theorem msc 26A06 ProofOfLeastAndReatestValueOfFunction LeastAndGreatestValueOfFunction