We start with a geometric definition:

Fig. 1 illustrates the definition and shows a curve, its point of inflection and the inflectional tangent (i.e. the tangent at the point of inflection) which crosses the curve. We can see that the point of inflection occurs where the curve stops bending to the left and starts bending to the right. This leads to a more descriptive definition of a point of inflection as a point where the direction of bending of a smooth curve changes sense.