[Math] How to determine the points of inflection for $f(x) = \frac{e^x}{1+e^x}$

Question

$$f(x) =\dfrac{e^x}{1+e^x}$$
I know we can find points of inflection using the second derivative test. The second derivative for the function above is $$f''(x) = \dfrac{e^x(1-e^x)}{(e^x+1)^3}$$ I have found one critical point for the second derivative which is $0$. I then determined that the function is concave up from $(-\infty,0)$ and concave down from $(0,\infty)$. I am now asked to find the points of inflection. How would I determine the exact points from where the function switches from concave up to concave down?

Answer 1

You've found the inflection point by identifying the value of $x$ at which the graph shifts from concave up, to concave down. Plus, it matches the solution to the $f''(x) = 0$.

That gives you an inflection point at $\left(0, \frac 12\right)$,