A triangle is drawn in a region bounded by the $x$-axis and a polynomial curve with all real roots, with one side of the triangle on the $x$-axis, as shown in this example:

[![enter image description here][1]][1]

>What is the supremum of the ratio of the triangle's area to the region's area?

(We require that the polynomial has all real roots, because [otherwise][2] the answer would be trivial: the supremum would be $1$, for example with $y=x-x^n$ and letting $n\to\infty$.)

I considered polynomials of the form $y=(-1)^{n-1}(x^2-1)(x^2-a^2)^n, a>1$, from $x=-1$ to $x=1$.

[![enter image description here][3]][3]

It seems that for any given $a$ value, the ratio $R$ is maximized with a certain $n$ value.  

For example, when $a=5$, the ratio $R$ is maximized when $n=47$:

$R\approx0.894915$ with $y=-(x^2-1)(x^2-5^2)^{\color{red}{46}}$

$R\approx0.894934$ with $y=(x^2-1)(x^2-5^2)^{\color{red}{47}}$

$R\approx0.894933$ with $y=-(x^2-1)(x^2-5^2)^{\color{red}{48}}$

I conjecture that the supremum of the ratio is approached when $a\to\infty$ (and for each value of $a$, we choose the value of $n$ that maximizes the ratio).


  [1]: https://i.sstatic.net/JWnw4.png
  [2]: https://math.stackexchange.com/q/4648043/398708
  [3]: https://i.sstatic.net/4Smma.png