Is $y=x^2$ smooth at origin?

Question

if $r(t)=t^2 i + t^4 j$, then it is a parabola $y = x^2$. It satisfies the condition of non smooth curve i.e. $\frac{dr}{dt}=0$ at $t=0$. But geometrically it shows the curve (parabola) is smooth at $(0,0)$. Why is this so?

Answer 1

A curve is non-smooth at a point if every parametrisation is either not differentiable or has $\frac{dr}{dt} = 0$ at that point. It's not enough that you have found one such parametrisation.

There are other parametisations of the same curve, such as $r(t) = ti + t^2j$, which does have a well-defined, non-zero derivative everywhere, showing that the curve is smooth.