I have a real world physical problem where I'm trying to determine the set of all possible coefficients of a function given specific constraints on the domain. I've managed to formulate the problem in mathematical terms, but I'm at a loss on how to proceed solving such a problem, or even defining what field of mathematics (other than perhaps simple algebra) might be used to solve the problem.
So what approach would I use to solve the following problem?
My first thoughts are that I have two (conditional) equations in two unknowns ($K_1 and K_2$), but the equations are nonlinear.
Let $f$ on $[0,\infty)$ be given by
$$f(x)=ax^2+bx$$ for all $x\geq 0$, where $a>0$ and $b\geq 0$.
If you want the image of $[x_0,x_1]$ to be a subset of $[y_0,y_1]$ then you need
$$f(x_0)=x_0^2a+x_0b\geq y_0, \tag{0}$$ $$f(x_1)=x_1^2a+x_1b\leq y_1.\tag{1}$$
Multiplying $(0)$ by $x_1$ and $(1)$ by $x_0$ and taking the difference gives
$$x_0x_1(x_1-x_0)a\leq x_0y_1-x_1y_0$$
so we need
$$a\leq \frac{x_0y_1-x_1y_0}{x_0x_1(x_1-x_0)}$$
Multiplying $(0)$ by $x_1^2$ and $(1)$ by $x_0^2$ and taking the difference gives
$$x_0x_1(x_0-x_1)b\leq x_0^2y_1-x_1^2y_0$$
so we need
$$b\geq -\frac{x_0^2y_1-x_1^2y_0}{x_0x_1(x_1-x_0)}$$
Let $X_i$ be the element mapped to $y_i$, i.e. $f(X_i)=y_i$. If you want the inverse image of $[y_0,y_1]$, i.e. $[X_0,X_1]$, to be a subset of $[x_0,x_1]$ then you want
$$x_0\leq X_0$$ $$x_1\geq X_1.$$
Since $f$ is increasing, these inequalities are equivalent to
$$ f(x_0)\leq f(X_0)=y_0$$ $$f(x_1)\geq f(X_1)=y_1.$$
i.e.
$$x_0^2a+x_0b\leq y_0 $$ $$x_1^2a+x_1b\geq y_1 $$
so just reverse the inequalities on $a$ and $b$ from the other case.
Note that it is not always possible to find such $a>0$ and $b\geq 0$. For example, if $x_0y_1\leq x_1y_0$, then in the first case we would need $a\leq0$, but $a>0$. Similarly, if $x_0^2y_1>x_1^2y_0$, then in the second case we would need $b< 0$, but $b\geq 0$.