Maxima, Minima and Points of Inflection

We often encounter situations in the physical sciences where we need to establish at which value(s) of an independent variable a maximum or minimum value in the function occurs. For example  

In general, y — f x) will display a number of turning points within the domain of the function. 

Turning points corresponding to maxima and minima may be classified as either  

Note that A is both a point of inflection and a stationary point, but while B and D are both points of inflection, they are not stationary points because f (x) 0. 

Points of inflection occur when the gradient is a maximum or minimum. This requires that f 2 x) = 0, but this in itself is not sufficient to characterize a point of inflection. We achieve this through the first nonzero higher derivative. 

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Understand the significance of higher-order derivatives and identify maxima, minima and points of inflection... 

This combination is best suited for small computer systems with realtime calculations, since only a few characteristic points must be stored for the peak separation. These points are defined by the first and second derivatives (maxima, minima and point of inflection) or can be easily determined during real-time processing (tangent point or slice point). 

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