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# 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 $f^{{\prime}}(b)\leq 0$.

Related:

ProofOfLeastAndReatestValueOfFunction, LeastAndGreatestValueOfFunction

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

26A06*no label found*

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