Dear Mir

First derivative need not to be zero.Pint of inflection can be identified where there is change of curvature.Since the first derivative is at an extremum, it follows that the second derivative, f″(x), is equal to zero if it exists, 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.

All the best.

AKASH GOYAL

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how to identify inflection pt if second derivative is not defined, without drawing the graph.

ex: the function 2x²/(x² – 1). It is concave up when |x| > 1 and concave down when |x| < 1. However, it has no points of     inflection because 1 and -1 are not in the domain of the function.