Umkehrpunkte

Umkehrpunkte, literally turning points, is a term used in German to denote points on a curve, function or trajectory where the direction of movement reverses. In mathematics, it commonly refers to points where a function changes its monotonicity: from increasing to decreasing or from decreasing to increasing. A turning point on an interval is typically a critical point x0 with f'(x0) = 0 or where f' is undefined, and the sign of f' changes around x0.

In the calculus context, turning points are closely related to local extrema. If at a critical point x0 the function changes from increasing to decreasing, x0 is a local maximum; if it changes from decreasing to increasing, x0 is a local minimum. The second derivative test provides a practical criterion: if f''(x0) > 0, x0 is a local minimum; if f''(x0) < 0, x0 is a local maximum; if f''(x0) = 0, the test is inconclusive and other methods may be needed. It is important to note that not every critical point is a turning point; for example, the function f(x) = x^3 has f'(0) = 0 but is increasing on both sides of 0, so 0 is not a turning point in the sense of a reversal of monotonicity.

In physics and geometry, a turning point describes where a motion reverses direction, such as a projectile reaching its maximum height (velocity zero) or a bound orbital motion reversing its course. The term is also used in applied settings to denote reversal points in dynamical systems or parametric curves, with attention to whether the trajectory actually changes direction rather than merely having a stationary slope.