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For $x\in[0,6]$, find the minimum of $m=x+\dfrac{96-16x}{3\sqrt{x^2+16}}$.

I used derivatives and found that the $x$ which minimizes $m$ is the positive root of \[9x^6+432x^4-2304x^2-49152x-28672=0.\] The approximate value is $x\approx4.71183$, $m\approx5.82339$. But I want the exact value. Since the polynomial function of degree $6$ has no formula to solve its roots, we need another method.

I can think of some alternative methods like using known inequalities or substituting to get rid of the square root, but I can't find any approaches using these methods currently.

  • Substitution $x=4 \tan y$ can help to get rid of the square root but it does not make it easier to find the exact minimum. – Vasili Feb 02 '23 at 13:43
  • This may be helpful: https://math.stackexchange.com/questions/835979/is-a-polynomial-equation-of-degree-ge-5-not-solvable-by-any-way?noredirect=1 I guess anyone with Mathematica could help. – Bumblebee Feb 02 '23 at 13:55
  • @Ajay I'm thinking that even if $x$ cannot be written as square root but the minimum can. –  Feb 02 '23 at 14:44
  • x can be written in terms of m though, through an ugly quartic formula. So if m has a closed form, x probably would too, through it is possible that symbolic solvers miss them – Phobo Havuz Feb 02 '23 at 15:06
  • Using Sage I got the result RuntimeError: no explicit roots found. Not much encouraging… Numerically: $x_{\min}\approx4.7118293977$ – jp boucheron Feb 02 '23 at 16:46
  • @youthdoo here’s an idea: in your working you may have come across this equation: $$1 - \frac{(-16+96x)x}{3(x^2 +16)^{3/2}}- \frac{16}{3 \sqrt{x^2 + 16}}=0.$$ Remember that for a fraction to be zero, the numerator must be zero. So if we get everything on the LHS in a single fraction, we can obtain something less than a degree 6. – Bumblebee Feb 02 '23 at 22:40
  • First, why you assume there is an exact solution? – CroCo Feb 06 '23 at 12:30

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