How to show $7^{th}$ degree polynomial is non-positive in $[0,1]$












4












$begingroup$



Let $0le xle 1$, show that inequality
$$99x^7-381x^6+225x^5-415x^4+157x^3-3x^2-x-1le 0$$




This problem comes from the fact that I solved a different inequality.I tried to solve it by factorizing it to see if I could get symbols.but I failed.



this inequality is hold by wolfapha Test it.






inequality polynomials a.m.-g.m.-inequality







share|cite|improve this question











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  • $begingroup$
    Not sure this helps, but here's what the part closest to the $x$-axis looks like for $x < 1$: wolframalpha.com/input/…
    $endgroup$
    – David K
    Dec 19 '18 at 12:44








  • 1




    $begingroup$
    How about using Sturm's Theorem en.wikipedia.org/wiki/Sturm%27s_theorem and show that the polynomial does not have real roots in the interval (0,1]?
    $endgroup$
    – Johannes Huisman
    Dec 19 '18 at 12:59
















4












$begingroup$



Let $0le xle 1$, show that inequality
$$99x^7-381x^6+225x^5-415x^4+157x^3-3x^2-x-1le 0$$




This problem comes from the fact that I solved a different inequality.I tried to solve it by factorizing it to see if I could get symbols.but I failed.



this inequality is hold by wolfapha Test it.






inequality polynomials a.m.-g.m.-inequality







share|cite|improve this question











$endgroup$












  • $begingroup$
    Not sure this helps, but here's what the part closest to the $x$-axis looks like for $x < 1$: wolframalpha.com/input/…
    $endgroup$
    – David K
    Dec 19 '18 at 12:44








  • 1




    $begingroup$
    How about using Sturm's Theorem en.wikipedia.org/wiki/Sturm%27s_theorem and show that the polynomial does not have real roots in the interval (0,1]?
    $endgroup$
    – Johannes Huisman
    Dec 19 '18 at 12:59














4












4








4


3



$begingroup$



Let $0le xle 1$, show that inequality
$$99x^7-381x^6+225x^5-415x^4+157x^3-3x^2-x-1le 0$$




This problem comes from the fact that I solved a different inequality.I tried to solve it by factorizing it to see if I could get symbols.but I failed.



this inequality is hold by wolfapha Test it.






inequality polynomials a.m.-g.m.-inequality







share|cite|improve this question











$endgroup$





Let $0le xle 1$, show that inequality
$$99x^7-381x^6+225x^5-415x^4+157x^3-3x^2-x-1le 0$$




This problem comes from the fact that I solved a different inequality.I tried to solve it by factorizing it to see if I could get symbols.but I failed.



this inequality is hold by wolfapha Test it.






inequality polynomials a.m.-g.m.-inequality




inequality polynomials a.m.-g.m.-inequality






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 19 '18 at 13:12









Michael Rozenberg

109k1896201




109k1896201










asked Dec 19 '18 at 12:12









inequalityinequality

777522




777522












  • $begingroup$
    Not sure this helps, but here's what the part closest to the $x$-axis looks like for $x < 1$: wolframalpha.com/input/…
    $endgroup$
    – David K
    Dec 19 '18 at 12:44








  • 1




    $begingroup$
    How about using Sturm's Theorem en.wikipedia.org/wiki/Sturm%27s_theorem and show that the polynomial does not have real roots in the interval (0,1]?
    $endgroup$
    – Johannes Huisman
    Dec 19 '18 at 12:59


















  • $begingroup$
    Not sure this helps, but here's what the part closest to the $x$-axis looks like for $x < 1$: wolframalpha.com/input/…
    $endgroup$
    – David K
    Dec 19 '18 at 12:44








  • 1




    $begingroup$
    How about using Sturm's Theorem en.wikipedia.org/wiki/Sturm%27s_theorem and show that the polynomial does not have real roots in the interval (0,1]?
    $endgroup$
    – Johannes Huisman
    Dec 19 '18 at 12:59
















$begingroup$
Not sure this helps, but here's what the part closest to the $x$-axis looks like for $x < 1$: wolframalpha.com/input/…
$endgroup$
– David K
Dec 19 '18 at 12:44






$begingroup$
Not sure this helps, but here's what the part closest to the $x$-axis looks like for $x < 1$: wolframalpha.com/input/…
$endgroup$
– David K
Dec 19 '18 at 12:44






1




1




$begingroup$
How about using Sturm's Theorem en.wikipedia.org/wiki/Sturm%27s_theorem and show that the polynomial does not have real roots in the interval (0,1]?
$endgroup$
– Johannes Huisman
Dec 19 '18 at 12:59




$begingroup$
How about using Sturm's Theorem en.wikipedia.org/wiki/Sturm%27s_theorem and show that the polynomial does not have real roots in the interval (0,1]?
$endgroup$
– Johannes Huisman
Dec 19 '18 at 12:59










1 Answer
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Because by AM-GM $$1+x+3x^2-157x^3+415x^4-225x^5+381x^6-99x^7=$$
$$=1+x+3x^2-157x^3+370x^4+x^4(45-225x+282x^2)+99x^6(1-x)geq$$
$$geq370x^4+3x^2+x+1-157x^3=6cdotfrac{185}{3}x^4+3x^2+x+1-157x^3ge$$
$$geq9sqrt[9]{left(frac{185}{3}x^4right)^6cdot3x^2cdot xcdot1}-157x^3=left(9sqrt[9]{left(frac{185}{3}right)^6cdot3}-157right)x^3geq0.$$






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    $begingroup$

    Because by AM-GM $$1+x+3x^2-157x^3+415x^4-225x^5+381x^6-99x^7=$$
    $$=1+x+3x^2-157x^3+370x^4+x^4(45-225x+282x^2)+99x^6(1-x)geq$$
    $$geq370x^4+3x^2+x+1-157x^3=6cdotfrac{185}{3}x^4+3x^2+x+1-157x^3ge$$
    $$geq9sqrt[9]{left(frac{185}{3}x^4right)^6cdot3x^2cdot xcdot1}-157x^3=left(9sqrt[9]{left(frac{185}{3}right)^6cdot3}-157right)x^3geq0.$$






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      5












      $begingroup$

      Because by AM-GM $$1+x+3x^2-157x^3+415x^4-225x^5+381x^6-99x^7=$$
      $$=1+x+3x^2-157x^3+370x^4+x^4(45-225x+282x^2)+99x^6(1-x)geq$$
      $$geq370x^4+3x^2+x+1-157x^3=6cdotfrac{185}{3}x^4+3x^2+x+1-157x^3ge$$
      $$geq9sqrt[9]{left(frac{185}{3}x^4right)^6cdot3x^2cdot xcdot1}-157x^3=left(9sqrt[9]{left(frac{185}{3}right)^6cdot3}-157right)x^3geq0.$$






      share|cite|improve this answer











      $endgroup$
















        5












        5








        5





        $begingroup$

        Because by AM-GM $$1+x+3x^2-157x^3+415x^4-225x^5+381x^6-99x^7=$$
        $$=1+x+3x^2-157x^3+370x^4+x^4(45-225x+282x^2)+99x^6(1-x)geq$$
        $$geq370x^4+3x^2+x+1-157x^3=6cdotfrac{185}{3}x^4+3x^2+x+1-157x^3ge$$
        $$geq9sqrt[9]{left(frac{185}{3}x^4right)^6cdot3x^2cdot xcdot1}-157x^3=left(9sqrt[9]{left(frac{185}{3}right)^6cdot3}-157right)x^3geq0.$$






        share|cite|improve this answer











        $endgroup$



        Because by AM-GM $$1+x+3x^2-157x^3+415x^4-225x^5+381x^6-99x^7=$$
        $$=1+x+3x^2-157x^3+370x^4+x^4(45-225x+282x^2)+99x^6(1-x)geq$$
        $$geq370x^4+3x^2+x+1-157x^3=6cdotfrac{185}{3}x^4+3x^2+x+1-157x^3ge$$
        $$geq9sqrt[9]{left(frac{185}{3}x^4right)^6cdot3x^2cdot xcdot1}-157x^3=left(9sqrt[9]{left(frac{185}{3}right)^6cdot3}-157right)x^3geq0.$$







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 19 '18 at 13:08

























        answered Dec 19 '18 at 13:01









        Michael RozenbergMichael Rozenberg

        109k1896201




        109k1896201






























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