Polytime and polyspace algorithm for determining the leading intersection of n discrete monotonic functions
Determining whether two monotone functions given as programs intersect is non-computable. Similarly, determining the first intersection under the promise that it exists is "arbitrarily hard" (definitely not polytime).

Given a program $P$, define a function $f_P$ which, for input $n$, is $1$ if $P$ stops after $n$ steps or less. The first intersection of $f_P$ and the constant function $1$ is the running time of $P$, if $P$ halts. So no program can decide whether $f_P$ and $1$ intersect.

Similarly, the time hierarchy theorem shows that for no recursive time bound $T$, the first intersection point can be found in time $T$, even under the promise that it exists. Using the space hierarchy theorem, you can get the same for space.