Suppose there is an item that deteriorates with use, its condition expressed as a number between $0$ (completely broken condition) and $1$ (perfect condition), for which I want to assign a monetary value. To make the units nicer, let's think of the condition also as a monetary value; i.e., someone has guaranteed to pay me $T$ dollars if I want to sell the item at any time, where $T$ is its condition. $M(T)$ is an implicit multiplicative markup on $T$ (i.e., $M(T) = 2$ would mean I value the item at twice the guaranteed resale value), and my overall value is $V(T) = M(T) * T$.
The first intuition (which I'm treating as an axiom) is that $V(T)$ should be monotonically increasing in $T$. The second is that $M(T)$ should be monotonically decreasing in $T$.
I can also estimate my $V$ at a handful of points in $T$. So far, I have tried Lagrange Polynomial Interpolation on such a handful of points to output a candidate $V(T)$, but when I divide this by $T$ to produce the corresponding $M(T)$, the result violates the second axiom; namely, polynomials produce non-decreasing regions in $M(T)$ around the LPI input points.
Are there ways to construct a $V(T)$, in elementary but perhaps non-polynomial functions, so that the two axioms are satisfied, and it also passes through a handful of input points (the points can be trusted to themselves satisfy the axioms)? Outside the regions corresponding to $0 \leq T \leq 1$, of course, the behavior of $V(T)$ and $M(T)$ do not matter. Is there a standard way to express valuations for these sorts of items?
