I have the following
$$y = \frac{x^2}{2}-\ln x$$
$$y'= x – \frac1x$$
I learned that inflection points were found by setting the $2^{nd}$ derivative equal to $0$, however, if I do that in this case I would get $i$, and I already checked and such is not possible in this case. However when I sent the $1^{st}$ derivative equal to $0$ I get,
$$x = \pm1$$
as possible inflection points which makes more sense.
Do I have a misconception as to how to find inflection points ? Or am I missing something ?
Best Answer
No. Points where the first derivative vanishes are called stationary points. If the second derivative exists (as it does in this case wherever the function is defined), it is a necessary condition for a point to be an inflection point that the second derivative vanishes. Thus the fact that there are no real solutions for the equation $y''=0$ shows that the function doesn't have any inflection points.