Proof That Normal Distribution has a Point of Inflexion One Standard Deviation From the Mean
If the random variableis described by a normal distributionthen the distribution ofis descibed by a probability density function
Differentiating this gives
Differentiating again (using the product rule) gives
Sinceif and only if
The points of inflexion are shown below (note that a point of inflexion is such that the curve crosses a tangent at a point).