Exponents and Triangle Sides

Let $f(x) = a^x + b^x - c^x$, where $c > a > 0$ and $c > b > 0$.

1. Let $M$ be the set of triples $(a, b, c)$ such that $a, b, c$ cannot be the side lengths of a triangle and $a = b$. Find the set of all possible zeros of $f(x)$ as $(a, b, c)$ ranges over $M$. 2. Suppose $a, b, c$ are the side lengths of a triangle $ABC$. Determine which of the following statements are correct, and justify your answers:

1. $f(x) > 0$ for all $x \in (-\infty, 1)$;
2. there exists $x \in \mathbb{R}$ such that $a^x, b^x, c^x$ cannot be the side lengths of a triangle;
3. if triangle $ABC$ is obtuse, then there exists $x \in (1, 2)$ such that $f(x) = 0$.