Existance of a fixed point of a bijective, smooth function:

Let $[a,b],,, [c,d]$ be two bounded closed intervals of $mathbb{R}$ such that $$[a,b]cap[c,d]not=emptyset$$ and $f:[a,b]to[c,d]$ be a bijective, smooth function.

We know that, if $[a,b]=[c,d],$ then $f$ has a fixed point.
My question is:

If $[a,b]not=[c,d],$ which condition(s) guaranteed the existence of a fixed point of $f$ ?