I came to be interested by the following rational function :
$$f(x)=\dfrac{(x^2-x+1)^3}{x^2(x-1)^2}\tag{1}$$
while writing this answer ; I discovered that $f$ is connected to rather deep features of "abstract algebra" (Klein $j$-invariant (see paragraph "sextic functions") ; as well( https://hsm.stackexchange.com/q/5038/3730) ; see also Luröth theorem.
Surprisingly, in this question and its second answer, one finds $f$ connected to a maximisation/minimization issue ; precisely, the largest value of constant $M$ such that :
$$(a^2+b^2+c^2-ab-bc-ca)^3 \geq M[((a-b)(b-c)(c-a)]^2\tag{2}$$
for all $a,b,c \geq 0$, is the minimum of values taken by $f(a)$.
Remark : This solution is based on the change of variables
$$(a,\ b, \ c) \rightarrow \ \ (ka+p, \ \ kb+p, \ \ kc+p)\ $$
Beyond this result and its short proof, is there a "higher level" rationale for a connection between expression (1) and maximisation problem (2) ?
The only feature I have noticed is that the RHS of (2) is the discriminant of polynomial $(x-a)(x-b)(x-c)$.
Another reference. Another one explaining what a j-invariant is Connection with elliptic curves
Take $x\in \Bbb{R}$, $a=x,b=1,c=0$, consider the polynomial $p(X)=X(X-1)(X-x)$. Find $d,A,B$ such that $4p(X+d)=4X^3-AX-B$ and compute the discriminant $Disc(p(X))=\frac{4A^3-B^2}{1728}$ and $j$-invariant $\frac{A^3}{Disc(p(X))}$ of the elliptic curve $E:Y^2=4X^3-AX-B$.
You'll find that the $j$-invariant is $256 f(x)$.
Then your $M$ is $\inf \frac{j(E)}{256}= \inf_{x\in \Bbb{R}}f(x)$.
Those relations give rise to useful modular functions $z\to\lambda(z),z\to j(z)$ where $\Im(z) > 0$ and $\Bbb{C}/(\Bbb{Z}+z\Bbb{Z})$ is a complex torus isomorphic to $E:Y^2=X(X-1)(X-\lambda(z))$ (ie. your $x$ is $\lambda(z)$ and your $f(x)$ is $j(z)$)