The following are all equivalent to the above definition:
Note that since the first derivative is at an extremum, it follows that the second derivative, f''(x), is equal to zero, but the latter condition does not provide a sufficient definition of a point of inflection. One also needs the lowest-order 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. (An example of such a function is ).
It follows from the definition that the sign of f'(x) on either side of the point (x,y) must be the same. If this is positive, the point is a rising point of inflection; if it is negative, the point is a falling point of inflection.
Points of inflection can also be categorised according to whether f'(x) is zero or not zero.
An example of a saddle point is the point (0,0) on the graph y=x³. The tangent is the x-axis, which cuts the graph at this point.
A non-stationary point of inflection can be visualised if the graph y=x³ is rotated slightly about the origin. The tangent at the origin still cuts the graph in two, but its gradient is non-zero.
Note that an inflection point is also called an ogee, although this term is sometimes applied to the entire curve which contains an inflection point.
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