Is this ratio of difference of these functions monotonic?

Question

1. If $f(x)$ is monotonic decreasing and $g(x)$ is monotonic increasing then is $([f(x)]^n - [g(x)]^n)/(f(x)-g(x))$ monotonicslly decreasing always or under any specific conditions? All functions above are real valued. I was thinking one condition would be if $g(x) < f(x)$. My intuition was also intending to show rate of decrease of numerator is greater than rate of decrease of denominator. Posting here to get a comprehensive and thorough answer...
2. If we instead had denominator to be just $f(x)$, what would be the answer then?

Answer 1

(I will assume that the denominators don't vanish). There are many examples where $\frac {f^{n}-g^{n}} {f-g}$ is not monotonic. If $f$ and $g$ are non-negative and monotonically increasing then $\frac {f^{n}-g^{n}} {f-g}$ is monotonically increasing using the comment by Herb Steinberg. If $f$ is decreasing and $g$ is increasing there is no reason for $\frac {f^{n}-g^{n}} {f-g}$ to be monotonic, even with $n=2$. Without non-negativity a simple counterexample with both $f$ and $g$ increasing is $f(x)=-1, g(x)=x$ with $n=3$. In this case $\frac {f^{n}-g^{n}} {f-g}$ is increasing for $x >-\frac 1 2$ and decreasing for $x <-\frac 1 2$. As for part 2), we can rarely say that the ratio $\frac {f^{n}-g^{n}} f$ is monotonic. Even with $n=1$ monotonicity may not hold.