Inflection Point

An Inflection Point is the point of a function in which its type of concavity changes—the function goes from concave to convex or from convex to concave.

Concavity and Convexity

• Concavity: the function is concave at a point if the second derivative f”(x) at that point is less than zero: f”(x) < 0
• Convexity: the function is convex at a point if the second derivative f”(x) at that point is greater than zero: f”(x) > 0

At a falling inflection point, the derivative has a local maximum; at a rising inflection point, it has a local minimum.

For an algebraic curve, a nonsingular point is an inflection point when the multiplicity of the intersection of the tangent line and the curve (at the point of tangency) is odd and greater than 2.

For a curve given by parametric equations, a point is an inflection point if its signed curvature changes from plus to minus or from minus to plus, i.e., changes sign.

For a twice differentiable function, a change in the sign of the second derivative across a point indicates a change in concavity. Finding a zero alone does not establish an inflection point.

Tipping Point

A tipping point is a threshold at which a system can shift sharply. It need not be a mathematical inflection point, though both terms are used more broadly to describe changes in a trend.

The tipping point is that magic moment when an idea, trend, or social behavior crosses a threshold, tips, and spreads like wildfire. —Malcolm Gladwell

Some systems with tipping points also exhibit hysteresis: their state depends on their history, and reversing the change may require crossing a different threshold.

Tipping points are commonly used in electric switches to assure fast opening and closing of switch contacts, to minimize electric arc formation and prevent burning or welding of the switch contacts.

In the climate system, a tipping point is a threshold that, when exceeded, can lead to large changes in the state of the system. Potential tipping points have been identified in the physical climate system and in impacted ecosystems. Feedback from the global carbon cycle is a driver for the transition between glacial and interglacial periods, with orbital forcing providing the initial trigger.

Examples

• At standard atmospheric pressure, water at 99 degrees Celsius is below its boiling point; at 100 degrees, added heat can drive boiling. The liquid does not all turn to vapor at once.
• A change in transmission can move a contagious process from declining to growing.
• Environmental thresholds can change which biological adaptations remain viable.
• Fishing pressure can push a population beyond its capacity to recover.
• Climate feedbacks can amplify warming after a threshold is crossed.
• Technology adoption can accelerate after enough users or supporting services are in place.