Two continuous, strictly decreasing, concave functions can intersect at most once?

Question

Suppose we have $f:R^{++} \rightarrow R$ and $g:R^{++} \rightarrow R$ s.t. both f and g are continuous functions. Further $f'(x)<0,g'(x)<0$ and $f''(x)>0,g''(x)>0$ for all x in their domains. Finally, f and g both tend towards infinity as x goes to 0 and both tend towards -infinity when x is large. My intuition is that f and g can cross at most once. However, I am having trouble ruling out the possibility that they cross twice. Any help would be greatly appreciated!

Answer 1

This can totally happen. You could start with such a function $f$ and then let $g$ be $f$ modified by very gentle oscillations, so gentle that they don't change the sign of the first or second derivative.

For instance, consider $f(x)=-\log x$ and $g(x)=-\log x+C\frac{\sin x}{1+x^3}$ for some constant $C>0$. We have $f'(x)=-1/x<0$ and $f''(x)=1/x^2>0$ for all $x>0$. The first and second derivative of $\frac{\sin x}{1+x^3}$ are nasty but it is easy to see that they are both bounded in absolute value by a constant multiple of $\frac{1}{1+x^3}$. If we choose $C$ small enough, then, this will not be enough to change the sign of $f'(x)$ or $f''(x)$, so $g'(x)$ is still negative and $g''(x)$ is still positive.

So, $f$ and $g$ satisfy your conditions. However, they cross infinitely many times, at every integer multiple of $\pi$.