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In Mathematics , particularly in Calculus , a stationary point is a Point on the Graph Of A Function where the Tangent to the graph is Parallel to the ''x''-axis (in 2D) or the plane tangent to the surface is parallel to the XY plane (in 3D). An equivalent definition is where the Derivative of the Function equals zero (known as a ''critical number''). An Inflection Point is a point where the Concavity changes. A point of inflection is not necessarily a stationary point. All inflection points have the property of ''f''''(''x'') = 0 but the reverse is not necessarily true. Stationary points of a real valued function ''f'': R → R are classified into four kinds:
Notice: Global (or absolute) maxima and minima are sometimes called global (or absolute) maximal (resp. minimal) extrema. While they may occur at stationary points, they are not actually an example of a stationary point. Determining the position and nature of stationary points aids in Curve Sketching , especially for Continuous Function s. Solving the equation ''f'''(''x'') = 0 returns the ''x''-coordinates of all stationary points; the ''y''-coordinates are trivially the function values at those ''x''-coordinates. The specific nature of a stationary point at ''x'' can in some cases be determined by examining the second derivative ''f''''(''x''):
A more straightforward way of determining the nature of a stationary point is by examining the function values between the stationary points. However, this is limited again in that it works only for functions that are continuous in at least a small interval surrounding the stationary point. A simple example of a point of inflection is the function ''f''(''x'') = ''x''3. There is a clear change of concavity about the point ''x'' = 0, and we can prove this by means of Calculus . The second derivative of ''f'' is the everywhere-continuous 6''x'', and at ''x'' = 0, ''f''′′ = 0, and the sign changes about this point. So ''x'' = 0 is a point of inflection. More generally, the stationary points of a real valued function ''f'': R''n'' → R are those points x0 where the derivative in every direction equals zero, or equivalently, the Gradient is zero. Example At x1 we have ''f' ''(''x'') = 0 and ''f''''(''x'') = 0. Even though ''f''''(''x'') = 0, this point is not a point of inflexion. The reason is that the sign of ''f' ''(''x'') changes from negative to positive. At x2, we have ''f' ''(''x'') 0 and ''f''''(''x'') = 0. But, x2 is not a stationary point, rather it is a point of inflexion. This because the concavity changes from concave upwards to concave downwards and the sign of ''f' ''(''x'') does not change; it stays positive. At x3 we have ''f' ''(''x'') = 0 and ''f''''(''x'') = 0. Here, x3 is both a stationary point and a point of inflexion. This is because the concavity changes from concave upwards to concave downwards and the sign of ''f' ''(''x'') does not change; it stays positive. SEE ALSO
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