I have a function $y=1/(1+e^{-x})$. I have been asked to use the first derivative to find any stationary points and then use the second derivative to classify them and provide points of inflection.
When I derive the function, I get the result $e^{-x}/(1+e^{-x})^2$. By setting this function to zero, I find that the equation is undefined at that point. Am I then correct to assume that the function contains no stationary points, and that therefore I cannot classify stationary points or find points of inflection? Or have I missed something?
Thanks.
Best Answer
The first derivative is
$$\frac{e^{-x}}{(1+e^{-x})^2},$$ which has no roots. Hence there are no stationary points.
The second derivative is
$$\frac{e^{-2x}-e^{-x}}{(1+e^{-x})^3},$$ which has a single root at $x=0$. Hence there is an inflection point.