Inflection point

In mathematics, a turning point is a point on a function graph, where the graph changes its curvature behavior: The graph changes here from either a right-hand to a left turn, or vice versa. This change is also called bow change. The determination of turning points is part of a curve sketching.

A turning point in the turning point occurs when the curvature of the graph changes its sign at the site. From this, several sufficient conditions for the determination of turning points derived. One criterion requires that the second derivative of the differentiable function changes its sign at the site. Other criteria require only that the second derivative of the function is zero, and that certain higher order derivatives are equal to zero.

If we consider the second derivative of a function as " increase its slope " can be their turning points as extreme points, ie maxima or minima, its slope ( its slope ) interpret.

Tangents through a turning point (in the picture painted red ) turning tangents called. Turning points where run this turn horizontal tangents are called saddle, terraces and horizontal turning points.

Definition

Be an open interval and a continuous function. It is said to have a turning point, and if there are intervals, so that either

• Is convex and concave in, or that
• Is concave and convex.

This clearly means that the graph of the function at the point changes the sign of its curvature. The curvature of a twice continuously differentiable function is described by its second derivative.

Criteria for the determination of turning points

In the following it is assumed that the function is sufficiently differentiable. Does not this, the following criteria in the search for turning points are not applicable. First, a necessary criterion is presented, that is, any twice continuously differentiable function must satisfy so that under certain circumstances there is a turning point at this point this criterion in one place. After some sufficient conditions are given. If these criteria are met, is safe from a turning point, but there are also turning points that do not meet these criteria sufficient.

Necessary criterion

Is a twice continuously differentiable function, then describes, as noted in the definition, the second derivative of the curvature of the function. Because an inflection point is a point at which the sign of the curvature, the second derivative of the function at that point must be zero. Thus:

Sufficient criterion without using the third derivative

When cornering discussions one of the following two sufficient conditions is usually used. In the first condition, only the second discharge occurs; for the sign of for and needs to be investigated.

Changes from negative to positive, then right-left turning point. If changes to from positive to negative, then a left-right reversal point.

Sufficient criterion, using the third derivative

In the second sufficient condition for a turning point, the third derivative is required, but only at the position itself, this condition is mainly used when the third derivative is easy to calculate. The main disadvantage compared to the already described condition is that in the event no decision can be made.

More of this and that a right-left turning point is. Accordingly, a left-right turning point for and.

Sufficient criterion using other derivatives

If the function is sufficiently differentiable, a decision can be made in the event:

This general formulation already contains the previous case. Beginning with the third derivative, the next discharge is sought, which is not zero. This is an odd derivative (based on the degree of extraction), there is a turning point.

Example

Then, the second derivative of the function is given by:

A turning point must be such that

. meet It follows. In order to clarify whether there is a genuine turning point at this point, you now also examines the third derivative:

From it can be concluded that there is a turning point. This fact can also be seen without using the third derivative: Due for and of itself changes the curvature behavior; therefore there must be a turning point.

The coordinate of this turning point is obtained by substituting in the functional equation.

The equation of the inflectional tangent can be determined by substituting the x coordinate of the bending point (2) in the first derivative. Thus obtained, the slope (m). Following are then added in the function definition (y = mx b) the determined x - & y -coordinate of the turning point and the m ( slope ) value. One then obtains the intersection with the y- axis (b) and therefore the complete equation of the inflectional tangent.

Special cases

The graph of the function changes in its curvature behavior (moving from right to left curve ). The first derivative at the point does not exist, the above formalism is therefore not applicable. Nevertheless, the function has at a turning point.

The graph of the function using the equation in the positive and in the negative range and, ie, Although it has a first but not a second discharge at the location, however there is a point of inflection before.