How the derivative might fail to exist Can a function have both a vertical tangent and cusp? Does The function $3x^{1/3}(x+2)$ have a vertical tangent and if so why? I believe that it has a cusp.

$y'(x) = \dfrac{x+2}{\sqrt[3]{x^2}} + 3\sqrt[3]x$, and $y'(0) = \text{undefined}$ .Thus there is a vertical tangent at $(0,0)$.