Inflexion

Inflexion is a term used in mathematics to describe a point on a curve where the curvature changes sign. This typically occurs at a local maximum or minimum of the second derivative of the function defining the curve. At an inflexion point, the curve transitions from being concave up to concave down, or vice versa. Geometrically, an inflexion point is where the tangent line to the curve crosses the curve itself.

To find inflexion points for a function f(x), one can analyze its second derivative, f''(x). If f''(x) = 0 at a point x=c, and the sign of f''(x) changes as x passes through c, then (c, f(c)) is an inflexion point. This sign change can be determined by examining the values of f''(x) for x slightly less than c and for x slightly greater than c. If f''(x) is continuous at c, the change in sign indicates an inflexion point. In some cases, an inflexion point may occur where the second derivative is undefined.

The concept of inflexion is important in various fields. In physics, it can represent a change in acceleration or a transition in a system's behavior. In economics, it might signify a turning point in growth rates. The term is also used in a more general sense to describe a point where a significant change or turning point occurs.