find_root does not fulfill tolerance

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brentq requires a sign change of $f$ in the given interval, but this cannot be achieved in $[-0.25, 0.4]$ (note that $f$ is tangent to $y=0$ at $0$). Maybe that's the reason?

Maybe - I see what you mean here. But some experimentation with some perturbations of your example don't give me confidence, since it doesn't satisfy its own documentation - http://sagecell.sagemath.org/?z=eJxtj...

..still i think this is a border case, because the root cannot be properly "enclosed". For completeness: it is also possible to do R.<x> = RDF[], f = 4*x^4 - x^2, f.roots().

Hmm, but RDF doesn't even do more than a certain tolerance. Maybe it only supports tol up to a certain point?

@kcrisman Yes, I think that's the case; some experiments with the function 4*x^4-x^2+0.01 around the exact root -1/2*sqrt(-1/10*sqrt(21) + 1/2) (this time in $[-0.25, 0.0]$) show it works ok until rtol is too small, giving ValueError: rtol too small (1e-16 < 4.44089e-16). But alternatively for the OP's question, let us now that using Newton's method, as in newton(4*x^4-x^2, -0.25, tol=1e-14, maxiter=3000), it will pass the tests with 1e-13 (and smaller).