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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 $a,b,c,d$?

Ross Millikan
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LCH
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  • Hint: When making a link, you can click on the chain in the edit window. That will give you a box to past the link into, then give you a chance to enter whatever descriptive text you like. That text will be blue and clicking will follow the link. – Ross Millikan May 08 '14 at 15:22

1 Answers1

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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: enter image description here

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 $u_2 \in ]0;\frac{-b\pm\sqrt{b^2-3ac}}{3a}[$

sirfoga
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  • Thank you so much for your detailed explanation and the elegant figures! I can understand easily. About the sign of $b$, it could be $b<0$ or $b>0$. On the other hand, it seems that under the condition $\Delta>0$, $c$ must satisfy $c<0$? – LCH May 08 '14 at 17:15
  • By the way, may I ask which software do you use for plotting this figure? – LCH May 08 '14 at 17:33
  • Simply download best of best Geogebra ! :-) – sirfoga May 08 '14 at 18:51
  • Thanks! I see. I've obtained information about Geogebra, and it seems very powerful and is easy to use:) – LCH May 08 '14 at 19:24