Qualitative inspection of solutions to $x^{4}-2x+1=0$ Consider the following polynomial
$$
x^{4}-2x+1=0
$$
Is it possible to check if there is or there is not a solution in $x\in\left]0,1\right[$ without explicitly evaluating the expression? What other tests are there to qualitatively classify the solutions for this polynomial?

$f(x) = x^4-2x+1$.
There are two sign changes, so there are $0$ or $2$ positive roots.
$f(-x)$ has no sign changes. So there are no negative roots.
Since $f(1)=0$, there are two positive roots.
Also $f(0)=1$.
Since $f'(x) = 4x^3 - 2$ is positive for all $x \ge 1$ (because $f''(x) = 12x^2 \ge 0$ implies that $f'$ is increasing and $f'(1)=2 > 0$), then $f(x)$ is positive for all $x > 1$.
So the second root must be somewhere between $x=0$ and $x=1$.
