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Indifferential calculus anddifferential geometry, aninflection point,point of inflection,flex, orinflection (rarelyinflexion) is a point on asmooth plane curve at which thecurvature changes sign. In particular, in the case of thegraph of a function, it is a point where the function changes from beingconcave (concave downward) toconvex (concave upward), or vice versa.
For the graph of a functionf ofdifferentiability classC2 (its first derivativef', and itssecond derivativef'', exist and are continuous), the conditionf'' =0 can also be used to find an inflection point since a point off'' = 0 must be passed to changef'' from a positive value (concave upward) to a negative value (concave downward) or vice versa asf'' is continuous; an inflection point of the curve is wheref'' =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 apoint of undulation orundulation point.
In algebraic geometry an inflection point is defined slightly more generally, as aregular point where the tangent meets the curve toorder at least 3, and an undulation point orhyperflex 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 thecurvature changes its sign.[2][3]
For example, the graph of thedifferentiable function has an inflection point at(x,f(x)) if and only if itsfirst derivativef' has anisolatedextremum atx. (This is not the same as saying thatf has an extremum). That is, in some neighborhood,x is the one and only point at whichf' has a (local) minimum or maximum. If allextrema off' areisolated, then an inflection point is a point on the graph off at which thetangent crosses the curve.
Afalling 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. Arising 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 byparametric equations, a point is an inflection point if itssigned curvature changes from plus to minus or from minus to plus, i.e., changessign.
For a smooth curve which is a graph of a twice differentiable function, an inflection point is a point on the graph at which thesecond derivative has an isolated zero and changes sign.
Inalgebraic geometry, a non singular point of analgebraic curve is aninflection point if and only if theintersection 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 analgebraic set. In fact, the set of the inflection points of a plane algebraic curve are exactly itsnon-singular points that are zeros of theHessian determinant of itsprojective completion.
For a functionf, if its second derivativef″(x) exists atx0 andx0 is an inflection point forf, thenf″(x0) = 0, but this condition is notsufficient 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 anundulation point. However, in algebraic geometry, both inflection points and undulation points are usually calledinflection points. An example of an undulation point isx = 0 for the functionf given byf(x) =x4.
In the preceding assertions, it is assumed thatf has some higher-order non-zero derivative atx, 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 off'(x) is the same on either side ofx in aneighborhood ofx. If this sign ispositive, the point is arising point of inflection; if it isnegative, the point is afalling point of inflection.
Points of inflection can also be categorized according to whetherf'(x) is zero or nonzero.
A stationary point of inflection is not alocal extremum. More generally, in the context offunctions of several real variables, a stationary point that is not a local extremum is called asaddle point.
An example of a stationary point of inflection is the point(0, 0) on the graph ofy =x3. The tangent is thex-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 ofy =x3 +ax, for any nonzeroa. The tangent at the origin is the liney =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 is concave for negativex and convex for positivex, 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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