Bounding distance between intersection point and minimizer of 2 monotonic functions

Question

Given a non decreasing $f:[0,1]\to\Bbb R_{\geq 0}$, and a non increasing function $g:[0,1]\to\Bbb R_{\geq 0}$ that intersect at $x^\perp$. And given $x^* = \operatorname{arg}\min_{0\leq x \leq 1}\{f(x)+g(x)\}$, can we bound $|x^\perp-x^*|$ ? What additional assumption would help bound the distance?