What Is an Inflection Point?

E.A. Sanker

The inflection point is an important concept in differential calculus. At the point of inflection, the curve of a function changes its concavity — in other words, it changes from negative to positive curvature, or vice versa. This point can be defined or visualized in different ways. In real-world applications where a system is being modeled using a curve, finding the inflection point is often critical in anticipating the system’s behavior.

Scientist with beakers
Scientist with beakers

Functions in calculus can be graphed on a plane consisting of an x and y axis, called a Cartesian plane. In any given function, the x value, or the value that is the input into the equation, produces an output, represented by the y value. When graphed, these values form a curve.

A curve can be either concave upwards or concave downwards, depending on the behavior of the function over certain values. A concave upwards region appears on a graph as a bowl-like curve opening upwards, while a concave downwards region opens downwards. The point at which this concavity changes is the inflection point.

There are a few different methods that may be helpful in visualizing where the inflection point lies on a curve. If one were to place a point on the curve with a straight line drawn through it that just touches the curve — a tangent line — and run that point along the course of the curve, the inflection point would occur at the exact point where the tangent line crosses over the curve.

Mathematically, the point of inflection is the point where the second derivative changes sign. The first derivative of a function measures the rate of change of a function as its input changes, and the second derivative measures how this rate of change itself may be changing. For example, a car’s velocity at a given moment is represented by the first derivative, but its acceleration — increasing or decreasing speed — is represented by the second derivative. If the car speeds up, its second derivative is positive, but at the point where it stops speeding up and begins to slow down, its acceleration and its second derivative become negative. This is the point of inflection.

To visualize this graphically, it is important to remember that the concavity of a function’s curve is expressed by its second derivative. A positive second derivative indicates a concave upward curve, and a negative second derivative indicates a curve that is concave downward. It is difficult to pinpoint the exact point of inflection on a graph, so for applications where it is necessary to know its exact value, the inflection point can be solved for mathematically.

One method of finding a function’s inflection point is to take its second derivative, set it equal to zero, and solve for x. Not every zero value in this method will be an inflection point, so it is necessary to test values on either side of x = 0 to make sure that the sign of the second derivative actually does change. If it does, the value at x is an inflection point.

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