At an inflection point, the second derivative is zero.
For example, consider cos(x) at x=π/2, since that's an inflection point. \[\frac{ d }{ dx }\cos x =-\sin x\]\[\frac{ d^{2} }{ dx ^{2} }\cos x =-\cos x\]\[-\cos \frac{ \pi }{ 2 } =0\]At that point, the first derivative, -sin x, is at a maximum, so the tangent at that point is a horizontal line. Since the first derivative is either at a maximum or a minimum at a point of inflection (on the original function), the second derivative will be zero at that same point.