For example, for the function $f(x) = x^{\frac{4}{3}}-4x^{\frac{1}{3}},$ there is a vertical tangent at $(0, 0),$ but it is also here the the concavity changes from negative to positive, as you can see from the second derivative. Is this point both a point of inflection and a vertical tangent?
My teacher's notes only state that it is a vertical tangent, which is confusing because I would have expected him to state that it is also a POI. Does being a vertical tangent imply that there is a POI?