This question is a follow up on a question (Show that $\{ x \in \mathbb R^2 : x_1x_2 = 1 \}$ is closed) that I asked earlier. The question is given by the following.
Given a point $\boldsymbol y = (y_1,y_2), y_1,y_2 \geq 0$, what is the minimum distance between this point and the curve $f(x) = 1/x$?
A natural way to model this would be to consider the distance function $f(x) = (x-y_1)^2 $ $+ (1/x - y_2)^2$, which gives the distance between $\boldsymbol y$ and the function $f(x)$ for some $x \in \mathbb R$. But finding an $x$ where the derivative $df(x)/dx$ is zero is equivalent to finding the root of a 4th degree polynomial... Anyone that has a suggestion on how to tackle this problem?
When some algebraic problem reduces to a general quartic equation, there cannot be any shortcut. Otherwise, you would have discovered a new way to a solve a quartic, and we have sufficient evidence that this is not possible.
We can exhibit cases such that four normals to an equilateral hyperbola are coincident, which shows that four extrema of the distance are possible. Hence, the problem is confirmed to be quartic and you can't simplify that.