Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Fermat's theorem (stationary points)
Summary
In mathematics, Fermat's theorem (also known as interior extremum theorem) is a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point (the function's derivative is zero at that point). Fermat's theorem is a theorem in real analysis, named after Pierre de Fermat.
By using Fermat's theorem, the potential extrema of a function , with derivative , are found by solving an equation in . Fermat's theorem gives only a necessary condition for extreme function values, as some stationary points are inflection points (not a maximum or minimum). The function's second derivative, if it exists, can sometimes be used to determine whether a stationary point is a maximum or minimum.
One way to state Fermat's theorem is that, if a function has a local extremum at some point and is differentiable there, then the function's derivative at that point must be zero. In precise mathematical language:
Let be a function and suppose that is a point where has a local extremum. If is differentiable at , then .
Another way to understand the theorem is via the contrapositive statement: if the derivative of a function at any point is not zero, then there is not a local extremum at that point. Formally:
If is differentiable at , and , then is not a local extremum of .
The global extrema of a function f on a domain A occur only at boundaries, non-differentiable points, and stationary points.
If is a global extremum of f, then one of the following is true:
boundary: is in the boundary of A
non-differentiable: f is not differentiable at
stationary point: is a stationary point of f
In higher dimensions, exactly the same statement holds; however, the proof is slightly more complicated. The complication is that in 1 dimension, one can either move left or right from a point, while in higher dimensions, one can move in many directions. Thus, if the derivative does not vanish, one must argue that there is some direction in which the function increases – and thus in the opposite direction the function decreases.
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Ontological neighbourhood