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How to show (but do not use numerical software such as Mathematica, Matlab...etc.) that this equation \begin{equation} \frac{u (83811 u-88223)+18076}{396-3276 u}-\frac{10 \log (u)}{3}-\frac{1}{2} \log \left(\frac{98 u+14}{273 u-33}\right)=0 \end{equation} has two roots $u_1$ and $u_2$ with $u_1,u_2>0$?

vonbrand
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LCH
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  • Why????????????????????? – Will Jagy May 08 '14 at 04:10
  • Because numerical evidences show that two positive roots exist, I would like to try proving this fact by using mathematical analysis. (Intermediate Value Theorem might be helpful?) – LCH May 08 '14 at 04:19

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Well, you can find the answers numerically first (but do not include in your write up). Make sure the function above satisfies intermediate-value theorem. And then just show there is a sign change around the values near the numerically determined solutions, and therefore there must be two solutions. Something like that should work.

user2485211
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  • What if the roots are of even multiplicity? – K1. May 08 '14 at 05:51
  • @Kevin: saying a root is of even multiplicity only makes sense for polynomials (although one could consider the exponent of the first non-zero term in the function's Taylor series around the zero, if it is differentiable) but besides, the answer provides a sufficient condition, not one that is strictly necessary. – Joshua Mundinger Jul 12 '14 at 04:59