Fermat’s theorem (stationary points)

Let $f:(a,b)\to ℝ$ 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 $f^{\prime }(a)\le 0$; if $f$ has a local minimum at $a$ then $f^{\prime }(a)\ge 0$. If $f$ is differentiable in $b$ and has a local maximum at $b$ then $f^{\prime }(b)\ge 0$ while if it has a local minimum at $b$ then $f^{\prime }(b)\le 0$.

Title	Fermat’s theorem (stationary points)
Canonical name	FermatsTheoremstationaryPoints
Date of creation	2013-03-22 13:45:05
Last modified on	2013-03-22 13:45:05
Owner	paolini (1187)
Last modified by	paolini (1187)
Numerical id	7
Author	paolini (1187)
Entry type	Theorem
Classification	msc 26A06
Related topic	ProofOfLeastAndReatestValueOfFunction
Related topic	LeastAndGreatestValueOfFunction