Let $a$, $b$, and $c$ be positive numbers. Consider the cubic equation $x^2(ax+b) = c$. Are there any useful bounds (upper and lower) I can put on the unique root of this equation for $x>0$? For example, we know that $x^* <(c/a)^{1/3}$ and $x^* < \sqrt{c/b}$. This equation can be solved explicitly, but that expression is rather complicated and I'm just looking to find an interval that contains a solution. Thanks!
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Note that, since your $f(x):=x^2(ax+b)-c$ is convex on $[0,+\infty)$ the Newton's iteration with initial point $x_0>0$ produces a sequence $x_n$ which is decreasing for $n\ge1$ and converges to $x^*$. From the equation, $y_n:=c/ax_n^2\ -\ b/a$ also converges to $x^*$, but increasing. This gives you an interval $[y_n, x_n]$ whose endpoints are rational functions of the coefficients (and as small as you wish, for large enough $n$). |
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