So far we dealt with stationary points of a function, meaning we looked for the roots of the first derivative. In case of inflection points, we look for the roots of the second derivative independently of the first.

When a function curves (is not a straight line) and the slope (the first derivative) changes, e.g. diminishes, graphically this is expressed as a clockwise (mathemetically negative) rotation if we follow the function in the direction of the independent variable (x). If the slope is growing, the rotation is counter-clockwise (positive mathematical sense). This graphical presentation is enhanced furthermore if we look at the animated motion of the tangent line. The point (if it exists) where this sense of rotation changes direction, is called an inflection point.

If the sense of rotation changes from clockwise(anticlockwise) to anticlockwise(clockwise), it means that the first derivative which is decreasing(increasing) starts to increase(decrease), and therefore the first derivative should have a minimum(maximum) at the inflection point. Of course the second derivative should vanish and the first consecutive non-zero derivative should be of odd-order. The sign of this derivative is related to the minimum or maximum of the first derivative. A root of the second derivative is a necessary, but not sufficient condition for an inflection point. A point of minimum or maximum of the first derivative is a necessary and sufficient condition for an inflection point.

The stationary point of y = x³ is also an inflection point. This and more examples are shown graphically at
Fig. Inflection points.

Inflection point rule

Look for vanishing second derivative. Among the consecutive derivatives look for the first non-vanishing. For an inflection point it should be of odd order.