Let $ \mathcal{B}= \{ f : [0, \infty) \rightarrow[0,1] \mid f(0) = 1, \text{ $f$ is a bijection and monotonically decreasing} \}$. Given $x_0 < x_1 \in [0, \infty)$ and $y_0 \in [0,1]$, I look for $$\arg\min_{f \in \mathcal{B}} f(x_1) \\ \text{such that } f(x_0) = y_0$$
In words, what is the function $f \in \mathcal{B}$ that "decreases the fastest" from $x_0$ to $x_1$? I am not sure how to approach it, and any help would be appreciated.