For which (a, b) the subset is not a smooth curve in the plane R^2? The full question is:
For which (a, b) the subset
C = {(x, y) ∈ $R^2 : y^2 = x^3 + ax + b$} ⊂ $R^2$
is not a smooth curve in the plane $R^2$?
I found out one similar question here:
How to check whether a subset of R^2 is a smooth curve
and have the beginning of the solution. First, I wrote the equation as $x^3 + ax + b - y^2 = 0$   and found the gradient
($3x^2+a; -2y$).
One of the answers in the link above was: "You are considering curves f(x,y)=C. By the implicit function theorem, you can find a smooth patch of the curve around any point where the gradient of the equation exists and is different from the zero vector."
Do I have to just check when the gradient is zero vector?
If yes, does that mean this?
$3x^2+a=0$
$-2y=0$
Then
$a=-3x^2$
$y=0$
Is it enough for answering the question? What about the parametr "b"? Can it be any number?
 A: Your solution is not enough yet. Define $f(x,y) := x^3+ax+b-y^2$. Then, by your two equations $3x^2+a=0$ and $-2y=0$, we have
$$
\text{grad}(f) = 0 \iff (x,y) = \left(\pm \sqrt{-\frac{a}{3}}, 0\right).
$$
Let's look at the case $(x,y) = (\sqrt{-a/3},0)$: we want to check in which cases this tuple lies in $C$. For this to hold, we must have that
\begin{equation}
0^2 = \left(\sqrt{-\frac{a}{3}}\right)^3 + a\sqrt{-\frac{a}{3}}+b,\tag1
\end{equation}
hence we can simply define $b$ as follows:
$$
b := -\left(\sqrt{-\frac{a}{3}}\right)^3 - a\sqrt{-\frac{a}{3}}.
$$
We immediately see that with this definition of $b$, equation (1) always holds, meaning that for all $a \leq 0$ the curve
$$
C_a := \left\{ (x,y) \in \mathbb{R}^2 : y^2 = x^3 +ax -\left(\sqrt{-\frac{a}{3}}\right)^3 - a\sqrt{-\frac{a}{3}} \right\}
$$
is not smooth. Indeed $a \leq 0$, since otherwise the square root in equation (1) were undefined. In conclusion, your curve is not smooth for all
$$
(a,b) = \left(a, -\left(\sqrt{-\frac{a}{3}}\right)^3 - a\sqrt{-\frac{a}{3}}\right),
$$
where $a \leq 0$. Here is a graphing calculator with your curve, so you can see that this solution works.
A similar argument works for the tuple $(x,y) = (-\sqrt{-a/3},0)$. I hope this is helpful!
