Difference between revisions of "Fermat theorem"
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| − | A necessary condition for a local extremum of a real-valued function. Suppose that a real-valued function | + | {{TEX|done}} |
| + | A necessary condition for a local extremum of a real-valued function. Suppose that a real-valued function $f$ is defined in a neighbourhood of a point $x_0\in\mathbf R$ and is differentiable at that point. If $f$ has a local extremum at $x_0$, then its derivative at $x_0$ is equal to zero: $f'(x_0)=0$. Geometrically this means that the tangent to the graph of $f$ at the point $(x_0,f(x_0))$ is horizontal. A condition equivalent to this for extrema of polynomials was first obtained by P. Fermat in 1629, but it was not published until 1679. | ||
Latest revision as of 13:01, 19 April 2014
A necessary condition for a local extremum of a real-valued function. Suppose that a real-valued function $f$ is defined in a neighbourhood of a point $x_0\in\mathbf R$ and is differentiable at that point. If $f$ has a local extremum at $x_0$, then its derivative at $x_0$ is equal to zero: $f'(x_0)=0$. Geometrically this means that the tangent to the graph of $f$ at the point $(x_0,f(x_0))$ is horizontal. A condition equivalent to this for extrema of polynomials was first obtained by P. Fermat in 1629, but it was not published until 1679.
Comments
For Fermat's theorems in number theory see Fermat great theorem; Fermat little theorem.
How to Cite This Entry:
Fermat theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fermat_theorem&oldid=18337
Fermat theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fermat_theorem&oldid=18337
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article