# Turning point

Turning point with tangent
Curvature behavior of the function sin (2x). The tangent is colored blue in convex areas, colored green in concave areas and colored red at turning points.

In mathematics , a turning point is a point on a function graph at which the graph changes its curvature behavior: Here the graph either changes from a right to a left curve or vice versa. This change is also called an arc change . The determination of turning points is part of a curve discussion .

A turning point at the turning point is when the curvature of the function graph changes its sign at the point . From this, various sufficient criteria for determining turning points can be derived. One criterion requires that the second derivative of the differentiable function changes its sign at this point . Other criteria only require that the second derivative of the function be zero and that certain higher derivatives are non-zero. ${\ displaystyle W \ left (x_ {W} | f (x_ {W}) \ right)}$ ${\ displaystyle \ x_ {W}}$${\ displaystyle x_ {W}}$${\ displaystyle f}$${\ displaystyle \ x_ {W}}$

If the second derivative of a function is viewed as the “slope of its slope”, its turning points can also be interpreted as [local] extreme points , that is, [local] maxima or minima, of its slope. ${\ displaystyle f}$

Tangents through a turning point (shown in red in the picture) are called turning tangents . Turning points at which these turning tangents run horizontally are called saddle, terrace or horizontal turning points .

Analogous to the term extreme value , the term reversal value for the corresponding function value seems intuitively plausible and is also used by some sources. However, it is pointed out directly or indirectly (through the use of quotation marks, for example ) that this is a rather unusual term. ${\ displaystyle f (x_ {W})}$

## definition

Be an open interval and a continuous function . It is said to have reached a turning point when there are intervals and so that either ${\ displaystyle {] a, b [} \ subset \ mathbb {R}}$${\ displaystyle f \ colon {] a, b [} \ to \ mathbb {R}}$${\ displaystyle f}$${\ displaystyle x_ {0}}$${\ displaystyle] \ alpha, x_ {0} [}$${\ displaystyle] x_ {0}, \ beta [}$

• ${\ displaystyle f}$is in convex and in concave , or that${\ displaystyle] \ alpha, x_ {0} [}$ ${\ displaystyle] x_ {0}, \ beta [}$
• ${\ displaystyle f}$is in concave and in convex.${\ displaystyle] \ alpha, x_ {0} [}$${\ displaystyle] x_ {0}, \ beta [}$

This clearly means that the graph of the function changes the sign of its curvature at the point . The curvature of a twice continuously differentiable function is described by its second derivative. ${\ displaystyle f}$${\ displaystyle x_ {0}}$

## Criteria for determining turning points

In the following it is assumed that the function can be differentiated sufficiently often. If this is not the case, the following criteria are not applicable to the search for turning points. First, a necessary criterion is presented, that is, every twice continuously differentiable function must meet this criterion at one point , so that there may be a turning point at this point. Then some sufficient criteria are given. If these criteria are met, there is certainly a turning point, but there are also turning points that do not meet these sufficient criteria. ${\ displaystyle f \ colon {] a, b [} \ to \ mathbb {R}}$${\ displaystyle x_ {W}}$

### Necessary criterion

Let be a twice continuously differentiable function, then, as already noted in the definition, the second derivative describes the curvature of the function graph. Since an inflection point is a point at which the sign of the curvature changes, the second derivative of the function must be zero at that point. The following applies: ${\ displaystyle f \ colon {] a, b [} \ to \ mathbb {R}}$${\ displaystyle f}$

If there is a turning point, it is .${\ displaystyle x_ {W}}$${\ displaystyle f \, '' (x_ {W}) = 0}$

### Sufficient criterion without using the third derivative

One of the following two sufficient conditions is usually used in curve discussions. In the first condition only the second derivative occurs; for this the sign of for and for must be examined. ${\ displaystyle f \, '' (x)}$${\ displaystyle x ${\ displaystyle x> x_ {W}}$

${\ displaystyle \ left. {\ begin {array} {ll} f {\ text {is two-fold differentiable in a neighborhood of}} x_ {W} {\ text {.}} \\ f \, '' (x) {\ text {changes at position}} x_ {W} {\ text {the sign.}} \ end {array}} \ right \} \ Rightarrow x_ {W} {\ text {is turning point.}}}$

If there is a change from negative to positive, it is a right-left turning point. When an changes from positive to negative, there is a left-right turning point. ${\ displaystyle \, f '' (x_ {W})}$${\ displaystyle x_ {W}}$${\ displaystyle \, f '' (x_ {W})}$${\ displaystyle x_ {W}}$${\ displaystyle x_ {W}}$

### Sufficient criterion using the third derivative

For the function f (x) = x 4 -x, the second derivative at x = 0 is zero; but (0,0) is not a turning point, since the third derivative is also zero and the fourth derivative is not zero.

In the second condition, which is sufficient for a turning point, the third derivative is also required, but only at the point itself. This condition is mainly used when the third derivative is easy to determine. The main disadvantage compared to the condition already explained is that in the case no decision can be made. ${\ displaystyle x_ {W}}$${\ displaystyle f \, '' '(x_ {W}) = 0}$

${\ displaystyle \ left. {\ begin {array} {ll} f {\ text {is in a neighborhood of}} x_ {W} {\ text {three times differentiable.}} \\ f \, '' (x_ { W}) = 0 \\ f \, '' '(x_ {W}) \ neq 0 \ end {array}} \ right \} \ Rightarrow x_ {W} {\ text {is turning point.}}}$

More precisely, it follows from and that at a minimum of the rise, that is, it has a right-left turning point, while conversely it has a left-right turning point for and at a maximum of the rise, that is to say a left-right turning point. ${\ displaystyle f \, '' (x_ {W}) = 0}$${\ displaystyle f \, '' '(x_ {W})> 0}$${\ displaystyle f}$${\ displaystyle x_ {W}}$${\ displaystyle f \, '' (x_ {W}) = 0}$${\ displaystyle f \, '' '(x_ {W}) <0}$${\ displaystyle x_ {W}}$

### Sufficient criterion using further derivations

If the function can be differentiated sufficiently often, a decision can also be made in the case . This is based on the expansion of at the point using the Taylor formula : ${\ displaystyle f}$${\ displaystyle f \, '' '(x_ {W}) = 0}$${\ displaystyle f}$${\ displaystyle x_ {0}}$

${\ displaystyle \ left. {\ begin {array} {ll} f {\ text {is in a neighborhood of}} x_ {W} \, n {\ text {times differentiable.}} \\ f \, ' '(x_ {W}) = \ ldots = f \, ^ {(n-1)} (x_ {W}) = 0 \\ f \, ^ {(n)} (x_ {W}) \ neq 0 \; {\ text {with}} \, n> 2 \, {\ text {and}} \, n \, {\ text {odd}} \ end {array}} \ right \} \ Rightarrow x_ {W } {\ text {is a turning point.}}}$

This more general formulation thus also contains the previous case: Starting with the third derivative, the next non-zero derivative is sought, and if this is a derivative of odd order, it is a turning point.

Or to put it quite generally: If the first non-zero derivative of the function at the point at which is is a derivative of odd order> 2, it has a turning point at this point. ${\ displaystyle f ^ {(n)}}$${\ displaystyle f}$${\ displaystyle x_ {0}}$${\ displaystyle f '' (x_ {0}) = 0}$${\ displaystyle f}$

## example

${\ displaystyle {f (x)} = {1 \ over 3} \ cdot x ^ {3} -2 \ cdot x ^ {2} +3 \ cdot x}$

Then the second derivative of the function is given by:

${\ displaystyle {f '' (x)} = {2 \ cdot x-4}}$

A turning point must be the condition ${\ displaystyle x_ {W}}$

${\ displaystyle {f '' (x)} = 0}$ or.
${\ displaystyle {2 \ cdot x-4} = 0}$

fulfill. It follows from this . In order to clarify whether there is actually a turning point at this point, we now also examine the third derivative: ${\ displaystyle x_ {W} = 2}$

${\ displaystyle {f '' '(x)} = 2 \,}$

From it can be concluded that this is a turning point. This fact can also be recognized without using the third derivative: Because of for and for , the curvature behavior changes; therefore there must be a turning point. ${\ displaystyle f \, '' '(x_ {W}) = f' '' (2) = 2 \ neq 0}$${\ displaystyle f \, '' (x) = 2 \ cdot x-4 <0}$${\ displaystyle x <2}$${\ displaystyle f \, '' (x) = 2 \ cdot x-4> 0}$${\ displaystyle x> 2}$

The coordinate of this turning point is obtained by inserting into the function equation . ${\ displaystyle y}$${\ displaystyle x = 2}$

${\ displaystyle y_ {W} = f (2) = {1 \ over 3} \ times 2 ^ {3} -2 \ times 2 ^ {2} +3 \ times 2 = {2 \ over 3}}$

The equation of the turning tangent can be determined by inserting the x-coordinate of the turning point ( 2 ) in the first derivative. This gives the slope (m). Then the determined x- & y-coordinate of the turning point and the m- (slope) value are inserted into the function determination ( y = mx + b ). You then get the point of intersection with the y-axis (b) and thus the complete equation of the turning tangent.

${\ displaystyle f \, '(x) = x ^ {2} -4 \ cdot x + 3}$
${\ displaystyle f \, '(2) = 2 ^ {2} -4 \ cdot 2 + 3 = -1}$
Turn tangent: ${\ displaystyle y = -x + {8 \ over 3}}$

## Special cases

The graph of the function changes its curvature behavior (transition from right to left curvature). The first derivation does not exist at this point , the above formalism is therefore not applicable. Still, the feature has reached a turning point. ${\ displaystyle f (x) = (x-2) \ cdot e ^ {| x |}}$${\ displaystyle x = 0}$${\ displaystyle x = 0}$${\ displaystyle x = 0}$

The graph of the function with the equation in the positive and negative regions and at , i.e. H. , has a first, but no second derivative at this point , but there is still a turning point. ${\ displaystyle f (x) = x ^ {2}}$${\ displaystyle f (x) = - x ^ {2}}$${\ displaystyle x = 0}$${\ displaystyle f (x) = x | x |}$${\ displaystyle 0}$

• Flat point, a point at which is (or at which is, but the curvature behavior does not change - depending on the definition)${\ displaystyle f '' = 0}$${\ displaystyle f '' = 0}$