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Fermat's theorem (stationary points) (Theorem)

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'(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$.



"Fermat's theorem (stationary points)" is owned by paolini.
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See Also: proof of least and greatest value of function, least and greatest value of function


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proof of Fermat's Theorem (stationary points) (Proof) by paolini
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Cross-references: local minimum, right derivative, local maximum, differentiable, extremum, continuous function
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This is version 4 of Fermat's theorem (stationary points), born on 2003-07-15, modified 2008-06-05.
Object id is 4450, canonical name is FermatsTheoremStationaryPoints.
Accessed 2892 times total.

Classification:
AMS MSC: 26A06 (Real functions :: Functions of one variable :: One-variable calculus)
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