Estimate on a positive root of a cubic equation

Question

Suppose that the cubic equation \begin{equation} a\,x^3+b\,x^2+c\,x+d=0, \end{equation} where $a,d>0$ and the discriminant $\Delta>0$. (refer to http://en.wikipedia.org/wiki/Cubic_function) ) Moreover, due to $\Delta>0$ the equation has three distinct real roots $u_1$, $u_2$ and $u_3$. Assume that $u_3<0$ and $u_1>u_2>0$. Then without using the solution formula for the cubic equation which appears so complicated, can we derive an estimate on $u_2$ (i.e. the smaller positive root) by means of the coefficients $a,b,c,d$?

Answer 1

You can restrict the interval in which look for $u_2$. In fact according to the given hypothesis the sketch must be similar to that one below:

There must be exactly 1 stationary point $\in ]u_3;u_2[$ and $\in ]u_2;u_1[$ (trivial proof). So $f'(x)=0$ for $x=\frac{-b\pm\sqrt{b^2-3ac}}{3a}$, depending on the sing of $b$, and thus $u_2 \in ]0;\frac{-b\pm\sqrt{b^2-3ac}}{3a}[$