Which point of inflection?

In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa.

For the graph of a function of differentiability class C2 (f, its first derivative f', and its second derivative f'', exist and are continuous), the condition f'' = 0 can also be used to find an inflection point since a point of f'' = 0 must be passed to change f'' from a positive value (concave upward) to a negative value (concave downward) or vice versa as f'' is continuous; an inflection point of the curve is where f'' = 0 and changes its sign at the point (from positive to negative or from negative to positive).[1] A point where the second derivative vanishes but does not change its sign is sometimes called a point of undulation or undulation point.

In algebraic geometry an inflection point is defined slightly more generally, as a regular point where the tangent meets the curve to order at least 3, and an undulation point or hyperflex is defined as a point where the tangent meets the curve to order at least 4.

Inflection points in differential geometry are the points of the curve where the curvature changes its sign.[2][3]

For example, the graph of the differentiable function has an inflection point at (x, f(x)) if and only if its first derivative f' has an isolated extremum at x. (this is not the same as saying that f has an extremum). That is, in some neighborhood, x is the one and only point at which f' has a (local) minimum or maximum. If all extrema of f' are isolated, then an inflection point is a point on the graph of f at which the tangent crosses the curve.

A falling point of inflection is an inflection point where the derivative is negative on both sides of the point; in other words, it is an inflection point near which the function is decreasing. A rising point of inflection is a point where the derivative is positive on both sides of the point; in other words, it is an inflection point near which the function is increasing.

For a smooth 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 smooth curve which is a graph of a twice differentiable function, an inflection point is a point on the graph at which the second derivative has an isolated zero and changes sign.

In algebraic geometry, a non singular point of an algebraic curve is an inflection point if and only if the intersection number of the tangent line and the curve (at the point of tangency) is greater than 2. The main motivation of this different definition, is that otherwise the set of the inflection points of a curve would not be an algebraic set. In fact, the set of the inflection points of a plane algebraic curve are exactly its non-singular points that are zeros of the Hessian determinant of its projective completion.

For a function f, if its second derivative f″(x) exists at x0 and x0 is an inflection point for f, then f″(x0) = 0, but this condition is not sufficient for having a point of inflection, even if derivatives of any order exist. In this case, one also needs the lowest-order (above the second) non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection, but an undulation point. However, in algebraic geometry, both inflection points and undulation points are usually called inflection points. An example of an undulation point is x = 0 for the function f given by f(x) = x4.

In the preceding assertions, it is assumed that f has some higher-order non-zero derivative at x, which is not necessarily the case. If it is the case, the condition that the first nonzero derivative has an odd order implies that the sign of f'(x) is the same on either side of x in a neighborhood of x. If this sign is positive, the point is a rising point of inflection; if it is negative, the point is a falling point of inflection.

Inflection points sufficient conditions:

Points of inflection can also be categorized according to whether f'(x) is zero or nonzero.

A stationary point of inflection is not a local extremum. More generally, in the context of functions of several real variables, a stationary point that is not a local extremum is called a saddle point.

An example of a stationary point of inflection is the point (0, 0) on the graph of y = x3. The tangent is the x-axis, which cuts the graph at this point.

An example of a non-stationary point of inflection is the point (0, 0) on the graph of y = x3 + ax, for any nonzero a. The tangent at the origin is the line y = ax, which cuts the graph at this point.

Some functions change concavity without having points of inflection. Instead, they can change concavity around vertical asymptotes or discontinuities. For example, the function x ↦ 1 x {\displaystyle x\mapsto {\frac {1}{x}}} is concave for negative x and convex for positive x, but it has no points of inflection because 0 is not in the domain of the function.

Some continuous functions have an inflection point even though the second derivative is never 0. For example, the cube root function is concave upward when x is negative, and concave downward when x is positive, but has no derivatives of any order at the origin.

If the function changes from positive to negative, or from negative to positive, at a specific point x = c, then that point is known as the point of inflection on a graph.

Both the concavity and convexity can occur in a function once or more than once. The point where the function is neither concave nor convex is known as inflection point or the point of inflection. In this article, the concept and meaning of inflection point, how to determine the inflection point graphically are explained in detail.

The point of inflection or inflection point is a point in which the concavity of the function changes. It means that the function changes from concave down to concave up or vice versa. In other words, the point in which the rate of change of slope from increasing to decreasing manner or vice versa is known as an inflection point. Those points are certainly not local maxima or minima. They are stationary points.

Generally, when the curve of a function bends, it forms a concave shape. It is known as the concavity of a function. In graph function, two types of concavity can be found.

Concave Up – If a curve opens in an upward direction or it bends up to make a shape like a cup, it is said to be concave up or convex down.

Concave Down – If a curve bends down or resembles a cap, it is known as concave down or convex up. In other words, the tangent lies underneath the curve if the slope of the tangent increases by the increase in an independent variable.

If f(x) is a differentiable function, then f(x) is said to be:

Here, f “(x) is the second order derivative of the function f(x).

The point of inflection defines the slope of a graph of a function in which the particular point is zero. The following graph shows the function has an inflection point.

It is noted that in a single curve or within the given interval of a function, there can be more than one point of inflection.

An inflection point is defined as a point on the curve in which the concavity changes. (i.e) sign of the curvature changes. We know that if f ” > 0, then the function is concave up and if f ” < 0, then the function is concave down. If the function changes from positive to negative, or from negative to positive, at a specific point x = c, then that point is known as the point of inflection on a graph.

We can identify the inflection point of a function based on the sign of the second derivative of the given function.  Also, by considering the value of the first-order derivative of the function, the point inflection can be categorized into two types, as given below.

If f'(x) is equal to zero, then the point is a stationary point of inflection.

If f'(x) is not equal to zero, then the point is a non-stationary point of inflection.

Click here to get the inflection point calculator.

Refer to the following problem to understand the concept of an inflection point.

Example:

Determine the inflection point for the given function f(x) = x4 – 24x2+11

Solution:

Given function: f(x) = x4 – 24x2+11

The first derivative of the function is

f’(x) = 4x3 – 48x

The second derivative of the function is

f”(x) = 12x2 – 48

Set f”(x) = 0,

12x2 – 48 = 0

Divide by 12 on both sides, we get

x2 – 4 = 0

x2 = 4

Therefore, x = ± 2

To check or x = 2, substitute x= 1 and 3 in f”(x)

So, f”(1) = 12(1)2 – 48 = -36 (negative)

f”(3) = 12(3)2 – 48 = 276 (positive)

To check for x = -2, substitute x= 0 and -3 in f”(x)

So, f”(0) = 12(0)2 – 48 = -48 (negative)

f”(3) = 12(3)2 – 48 = 276 (positive)

Hence, proved

Now, substitute x = ± 2 in f”(x)

Therefore, it becomes

f”(2) = 12(2)2 – 48 = -69

f”(-2) = 12(-2)2 – 48 = -69

Therefore, the inflection points are (2, -69), and (-2, -69).

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