Flat point

This article covers flat points of function graphs. For flat points of regular surfaces see main curvature

A turning point can be seen on the left of the picture, a (real) flat point on the right.

A flat point is a point on the graph of a real function at which the second derivative of the function (which can be differentiated at least twice at this point) = 0. Flat points therefore include both inflection points (at which the second derivative of has a change in sign , i.e. a left curve changes to a right curve or vice versa; see the point with the green tangent in the graphic ) and points at which the curve behavior does not change. The latter are sometimes referred to as real flat points (see the point with the blue tangent in the graphic ). In the case of real flat points, in addition to the second, (at least) the third derivative is also equal to 0 with a change in sign (provided that the function can be differentiated at least three times at this point). ${\displaystyle (x_{0}|f(x_{0}))}$ ${\displaystyle f}$ ${\displaystyle x=-1}$ ${\displaystyle x=2}$

If flat points have a rise, they can be differentiated into flat rise and flat fall according to their type .

Other definition

Some authors also require that the curvature behavior does not change and that the increase is not equal to zero; Flat points are then the zeros of the even order of the 2nd derivative. The flat points according to this alternative definition (for example the point with the blue tangent in the graphic ) would be exactly the real flat points according to the first definition. In particular, turning points (for example the point with the green tangent in the graphic ) are no flat points according to this definition . Extreme points with multiple zeros are then not flat points either . ${\displaystyle x=2}$ ${\displaystyle x=-1}$

The following then has the same meaning: and has a zero point of even multiplicity at . Then it has an extremum at this point , since there is no change in sign from at ; thus there is no change in the direction of curvature, and there is a flat point and no turning point . ${\displaystyle f''(x_{0})=0}$ ${\displaystyle f''}$ ${\displaystyle x_{0}}$ ${\displaystyle f''}$ ${\displaystyle x_{0}}$ ${\displaystyle f''}$ ${\displaystyle x_{0}}$

Individual evidence

1. ^ Mathematical formulas and definitions. Friedrich Barth a. a. (Editor), Bayerischer Schulbuch-Verlag, p. 64
2. ^ Elements of Mathematics 11/12 Lower Saxony, Schroedel Verlag, Braunschweig, 2009, ISBN 978-3-507-87920-1 , p. 29
3. ↑ flat points. In: mathenexus.zum.de. August 24, 2004, accessed January 9, 2015 . 
4. ↑ Practice of Mathematics . Volume 37, Aulis Verlag Deubner, 1995, p. 57 ( limited preview in Google book search)