Alon Moder

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). 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.

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.

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Presenters: Anne Gresham, RN, CNS; and Chris Kvidera, LICSW

Dialectical Behavior Therapy (DBT) was developed by Dr. Marsha Linehan, PhD of the University of Washington.   It is a research proven method for treating the most difficult clients;  i.e.: borderline personality disorder, co-occurring (dual diagnoses) disorders, and other personality disorders.  Current Research is being conducted all over the world with many client populations. The goal of DBT is the achievement of a quality life rather than the prevention of crises. If you would like to stop the “revolving door” pattern of crisis management with your clients, this training will help. As the individual therapist, you ARE the key to your client’s recovery.

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On Dec. 14, 2020, a board of enlisted leaders from Maxwell Air Force Base gathered, virtually, to conduct the quarterly senior airman below-the-zone board. However, something was different: all identifying information for the candidates had been removed.

Prior to the board meeting, Staff Sgt. Alexcia Tubbs and Senior Airman Micaella Trinidad Howard proposed the idea to “sanitize” the packages by removing names, race, ethnicity, unit and gender from the packages to eliminate any subconscious bias in the scoring process.

Their idea aligns with the recently formed 42nd ABW Freedom to Serve Initiative, which aims to identify and remove barriers from Airmen’s success and promote a more inclusive force. The Freedom to Serve team acknowledged the benefit of removing personal identifiers to ensure the board was entirely merit-based.

The new board process also aligns with the Chief of Staff of the Air Force Gen. Charles Q. Brown Jr.’s commitment to accelerating positive change in the Air Force. In his paper, “Accelerate Change or Lose”, Brown emphasizes the importance of empowering Airmen to be innovative problem-solvers and creating an environment in which all Airmen can reach their full potential, which is exactly what Tubbs and Trinidad Howard did.

“During my 3 years of experience working BTZ boards, I feel as though this was a necessary action,” said Tubbs. “I believe it needed to take place in order to create a fair panel and remove any biases.”

Trinidad Howard also believes the new board process is crucial to securing the freedom to serve without interference of bias.

“The ultimate goal of sanitizing BTZ packages was to remove bias in the decision making,” said Trinidad Howard. “It’s our responsibility to ensure that all airmen have a fair chance of being picked.”

To make things fair, you won't see Last Seen timestamps for people with whom you That is why even if the exact Last Seen time is hidden ,

The maximum takeoff mass (MTOM), often referred to as maximum takeoff weight (MTOW), of an aircraft is a value defined by the aircraft manufacturer It is the maximum mass at which the aircraft is certified for take off due to structural or other limits MTOW is usually specified in units of kilograms or pounds