I'm trying to prove the following:
Let $f,g:I\to I$ , where $I=[a,b]\subset\mathbb{R}$ is some compact interval, be $C^1$ orientation preserving interval diffeomorphisms each with exactly two fixed points, then $f, g$ are topologically conjugate
$\textbf{My attempt:}$ I know I need to show that both maps have exactly one attracting fixed point and one repelling fixed point. I was able to show that it has two fixed points using the intermediate value theorem and the definition of continuity, but I'm lost on how to show that one is attracting and the other repelling. After that I would need to construct fundamental domains but I'm still unsure how to do that and piece it all together.
Hint:
Consider the graph of the diffeomorphism as a part of a square $[a,b]^2$. The curve has to lie completely above the main diagonal, or completely below, except for the two endpoints which correspond to the fixed points $a$, $b$. In the first case (curve above the diagonal), $b$ is the attractive one, whereas in the second case $a$ is the attractive one.
Hint 2:
Take any point $x$ in your interval, that is not $a$ or $b$. Then the interval $[x, f(x))$ is a fundamental domain (in case $f(x)>x$; otherwise take $[f(x), x)$ instead).