Triangles on a Convex Exponential Curve

Let $f(x)=\mathrm e^x+x$. For three points $A,B,C$ on the curve $y=f(x)$ whose $x$-coordinates form an arithmetic progression, determine, with proof, which of the following are true.

1. $\triangle ABC$ is always obtuse. 2. $\triangle ABC$ can be right-angled. 3. $\triangle ABC$ can be isosceles. 4. $\triangle ABC$ cannot be isosceles.