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There exists a,b,c,d,e such that a^5-a^4 b+b^5-b^4 c+c^5-c^4 d+d^5-d^4 e-a e^4+e^5<5151/10000 (a b^4-a^2 b^2 c+b c^4-b^2 c^2 d+c d^4+a^4 e-c^2 d^2 e-a^2 b e^2-a d^2 e^2+d e^4)and a>0,b>0,c>0,d>0,e>0. How to get a set of solution of the inequality?

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2 Answers 2

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Try NMinimize

NMinimize[{1, 
a^5 - a^4 b + b^5 - b^4 c + c^5 - c^4 d + d^5 - d^4 e - a e^4 +e^5 < 5151/10000 (a b^4 - a^2 b^2 c + b c^4 - b^2 c^2 d + c d^4 + a^4 e - c^2 d^2 e - a^2 b e^2 - a d^2 e^2 + d e^4),a > 0, b > 0, c > 0, d > 0, e > 0}
, {a, b, c, d, e} ]
(*{1., {a -> 0.00100011, b -> 0.00100011, c -> 0.0155396, d -> 0.0140353, e -> 0.00100011}}*)
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  • $\begingroup$ sol=Last[NMinimize[{1,a^5-a^4 b+b^5-b^4 c+c^5-c^4 d+d^5-d^4 e-a e^4+e^5<5151/10000(a b^4-a^2 b^2 c+b c^4-b^2 c^2 d+c d^4+a^4 e-c^2 d^2 e-a^2 b e^2-a d^2 e^2+d e^4),a>0,b>0,c>0,d>0,e>0},{a,b,c,d,e}]]; a^5-a^4 b+b^5-b^4 c+c^5-c^4 d+d^5-d^4 e-a e^4+e^5<(5151 (a b^4-a^2 b^2 c+b c^4-b^2 c^2 d+c d^4+a^4 e-c^2 d^2 e-a^2 b e^2-a d^2 e^2+d e^4))/10000/.sol the result is False I think it may be caused by a very small error. $\endgroup$
    – lapcal
    Commented May 12, 2022 at 10:39
  • $\begingroup$ Try a^5-a^4 b+b^5-b^4 c+c^5-c^4 d+d^5-d^4 e-a e^4+e^5-5151/10000(a b^4-a^2 b^2 c+b c^4-b^2 c^2 d+c d^4+a^4 e-c^2 d^2 e-a^2 b e^2-a d^2 e^2+d e^4) (*2.79237*10^-10*) to see that it's only a numerical issue! $\endgroup$ Commented May 12, 2022 at 10:49
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divide the left-hand side by the right-hand side and minimize:

NMinimize[{(a^5 - a^4 b + b^5 - b^4 c + c^5 - c^4 d + d^5 - d^4 e - a e^4 + e^5)/
           (a b^4 - a^2 b^2 c + b c^4 - b^2 c^2 d + c d^4 + a^4 e - c^2 d^2 e - a^2 b e^2 - a d^2 e^2 + d e^4),
           a > 0 && b > 0 && c > 0 && d > 0 && e > 0}, {a, b, c, d, e}]

(*    {0.515091, {a -> 5.71488, b -> 5.68244, c -> 4.14814, d -> 1.13836, e -> 4.24523}}    *)

The minimized ratio is smaller than 5151/10000.

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  • $\begingroup$ There is a chance you divide a negative number by a negative number. In order to complete the answer, a b^4 - a^2 b^2 c + b c^4 - b^2 c^2 d + c d^4 + a^4 e - c^2 d^2 e - a^2 b e^2 - a d^2 e^2 + d e^4 /. {a -> 5.71488, b -> 5.68244, c -> 4.14814, d -> 1.13836, e -> 4.24523} returns 3966.18 $\endgroup$
    – user64494
    Commented May 12, 2022 at 12:09
  • $\begingroup$ Another adjustment is to add (a b^4 - a^2 b^2 c + b c^4 - b^2 c^2 d + c d^4 + a^4 e - c^2 d^2 e - a^2 b e^2 - a d^2 e^2 + d e^4)>0 to the restrictions. $\endgroup$
    – user64494
    Commented May 12, 2022 at 12:33

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