Counting Stationary Points

“No stationary points” means $f'(x)=0$ has no solutions in the domain. If the derivative is quadratic, use its discriminant: negative means no real roots, zero means one repeated root, positive means two distinct roots. These count stationary inputs, not roots of the original curve.

At $p=\pm3$ the derivative has a repeated root: one stationary point remains. These boundary values are excluded.

For which real values of $k$ does $y=x^3+kx^2+12x$ have two distinct stationary points?

The derivative is $3x^2+2kx+12$. Its discriminant must be positive: $4k^2-144>0$. Hence $k<-6$ or $k>6$.