Inflection points on elliptic curves

Question

I'm looking at how to transform a cubic into a Weierstrass equation on page 52 of this elliptic curves pdf here. The author writes: “There are two distinct transformations depending on whether the point $P$ is an inflection point on $C$ or not.” On page 53 he gives the point $P\left(1,1,1\right)$ on the cubic $$X^{3}+2Y^{3}-3Z^{3}=0$$as an example of a point that isn't an inflection point. Apart from doing the calculations and seeing that the tangent line at this point meets the curve again, how to tell that $P$ isn't an inflection point?