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How to prove the inequality  , assuming that the variables are nonnegative? That hard question was asked by arqady in dxdy and answered by himself in a complicated way. Maple proves the inequality by the LagrangeMultipliers command which is strong. I think these calculations cannot be done by hand at all. Without loss of generality one may assume  . Then
restart:with(Student[MultivariateCalculus]):
ans := [LagrangeMultipliers((a+b+c+d)*(a*b+a*c+a*d+b*c+b*d+c*d)-12*sqrt((a^2+b^2+c^2+d^2)*a*b*c*d), [a+b+c+d-1], [a, b, c, d], output = detailed)]:
We have to remove complex solutions by
ans1:=remove(c -> has(evalf(c), I),ans):
The next big output is only partly seen in the post (look in the attached file for the whole one).
ans2:=simplify(ans1,radical);
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[[a = 1/6, b = 1/2, c = 1/6, d = 1/6, lambda[1] = 0, -12*sqrt((a^2+b^2+c^2+d^2)*a*b*c*d)+(b+c+d)*a^2+(b^2+(3*c+3*d)*b+c^2+3*c*d+d^2)*a+(d+c)*b^2+(c^2+3*c*d+d^2)*b+c^2*d+c*d^2 = 0],[a = 1/4, b = 1/4, c = 1/4, d = 1/4, lambda[1] = 0, -12*sqrt((a^2+b^2+c^2+d^2)*a*b*c*d)+(b+c+d)*a^2+(b^2+(3*c+3*d)*b+c^2+3*c*d+d^2)*a+(d+c)*b^2+(c^2+3*c*d+d^2)*b+c^2*d+c*d^2 = 0],[a = 13/72-(1/216)*sqrt(3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))+(1/216)*sqrt(3)*sqrt(2)*sqrt((11*(11548+108*sqrt(13397))^(1/3)-(11548+108*sqrt(13397))^(2/3)+284)/(11548+108*sqrt(13397))^(1/3)+273*sqrt(3)/sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))), b = 11/24+(1/72)*sqrt(3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))-(1/72)*sqrt(3)*sqrt(2)*sqrt((11*(11548+108*sqrt(13397))^(1/3)-(11548+108*sqrt(13397))^(2/3)+284)/(11548+108*sqrt(13397))^(1/3)+273*sqrt(3)/sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))), c = 13/72-(1/216)*sqrt(3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))+(1/216)*sqrt(3)*sqrt(2)*sqrt((11*(11548+108*sqrt(13397))^(1/3)-(11548+108*sqrt(13397))^(2/3)+284)/(11548+108*sqrt(13397))^(1/3)+273*sqrt(3)/sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))), d = 13/72-(1/216)*sqrt(3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))+(1/216)*sqrt(3)*sqrt(2)*sqrt((11*(11548+108*sqrt(13397))^(1/3)-(11548+108*sqrt(13397))^(2/3)+284)/(11548+108*sqrt(13397))^(1/3)+273*sqrt(3)/sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))), lambda[1] = -(5/36)*(sqrt(2)*(sqrt(3)*(sqrt(13397)-(71/27)*(11548+108*sqrt(13397))^(1/3)-(103/540)*(11548+108*sqrt(13397))^(2/3)+2887/27)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))-15*sqrt(13397)+(355/9)*(11548+108*sqrt(13397))^(1/3)+(109/36)*(11548+108*sqrt(13397))^(2/3)-14435/9)*sqrt((11*(11548+108*sqrt(13397))^(1/3)-(11548+108*sqrt(13397))^(2/3)+284)/(11548+108*sqrt(13397))^(1/3)+273*sqrt(3)/sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3)))-(133/15)*(11548+108*sqrt(13397))^(2/3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))+(2*((sqrt(13397)+2374/45)*(11548+108*sqrt(13397))^(1/3)+(103/5)*sqrt(13397)+(449/90)*(11548+108*sqrt(13397))^(2/3)+132727/45))*sqrt(3))/((11548+108*sqrt(13397))^(2/3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))), -12*sqrt((a^2+b^2+c^2+d^2)*a*b*c*d)+(b+c+d)*a^2+(b^2+(3*c+3*d)*b+c^2+3*c*d+d^2)*a+(d+c)*b^2+(c^2+3*c*d+d^2)*b+c^2*d+c*d^2 = -(13/46656)*(((2/13)*sqrt((11*(11548+108*sqrt(13397))^(1/3)-(11548+108*sqrt(13397))^(2/3)+284)/(11548+108*sqrt(13397))^(1/3)+273*sqrt(3)/sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3)))*(11548+108*sqrt(13397))^(1/3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))+sqrt(2)*(sqrt(3)*(11548+108*sqrt(13397))^(1/3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))-(11/13)*(11548+108*sqrt(13397))^(1/3)-(2/13)*(11548+108*sqrt(13397))^(2/3)+568/13))*sqrt(5)*sqrt((sqrt(3)*sqrt(2)*sqrt((11*(11548+108*sqrt(13397))^(1/3)-(11548+108*sqrt(13397))^(2/3)+284)/(11548+108*sqrt(13397))^(1/3)+273*sqrt(3)/sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3)))-sqrt(3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))-33)*(sqrt(3)*sqrt(2)*sqrt((11*(11548+108*sqrt(13397))^(1/3)-(11548+108*sqrt(13397))^(2/3)+284)/(11548+108*sqrt(13397))^(1/3)+273*sqrt(3)/sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3)))-sqrt(3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))+39)*(sqrt(2)*(sqrt(3)*(11548+108*sqrt(13397))^(1/3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))+(11/5)*(11548+108*sqrt(13397))^(1/3)+(2/5)*(11548+108*sqrt(13397))^(2/3)-568/5)*sqrt((11*(11548+108*sqrt(13397))^(1/3)-(11548+108*sqrt(13397))^(2/3)+284)/(11548+108*sqrt(13397))^(1/3)+273*sqrt(3)/sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3)))-(216/5)*(11548+108*sqrt(13397))^(1/3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))-(328/5*((11548+108*sqrt(13397))^(1/3)+(5/164)*(11548+108*sqrt(13397))^(2/3)-355/41))*sqrt(3))/((11548+108*sqrt(13397))^(1/3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))))-(180/13)*sqrt(2)*(sqrt(3)*(11548+108*sqrt(13397))^(1/3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))+(11/5)*(11548+108*sqrt(13397))^(1/3)+(2/5)*(11548+108*sqrt(13397))^(2/3)-568/5)*sqrt((11*(11548+108*sqrt(13397))^(1/3)-(11548+108*sqrt(13397))^(2/3)+284)/(11548+108*sqrt(13397))^(1/3)+273*sqrt(3)/sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3)))-(15552/13)*(11548+108*sqrt(13397))^(1/3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3))+(11808/13*((11548+108*sqrt(13397))^(1/3)+(5/164)*(11548+108*sqrt(13397))^(2/3)-355/41))*sqrt(3))/((11548+108*sqrt(13397))^(1/3)*sqrt((2*(11548+108*sqrt(13397))^(2/3)+11*(11548+108*sqrt(13397))^(1/3)-568)/(11548+108*sqrt(13397))^(1/3)))]
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(1) |
evalf(ans2);
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![[[a = .1666666667, b = .5000000000, c = .1666666667, d = .1666666667, lambda[1] = 0., -12.*((a^2+b^2+c^2+d^2)*a*b*c*d)^(1/2)+(b+c+d)*a^2+(b^2+(3.*c+3.*d)*b+c^2+3.*c*d+d^2)*a+(d+c)*b^2+(c^2+3.*c*d+d^2)*b+c^2*d+c*d^2 = 0.], [a = .2500000000, b = .2500000000, c = .2500000000, d = .2500000000, lambda[1] = 0., -12.*((a^2+b^2+c^2+d^2)*a*b*c*d)^(1/2)+(b+c+d)*a^2+(b^2+(3.*c+3.*d)*b+c^2+3.*c*d+d^2)*a+(d+c)*b^2+(c^2+3.*c*d+d^2)*b+c^2*d+c*d^2 = 0.], [a = .1666666667, b = .1666666667, c = .5000000000, d = .1666666667, lambda[1] = 0., -12.*((a^2+b^2+c^2+d^2)*a*b*c*d)^(1/2)+(b+c+d)*a^2+(b^2+(3.*c+3.*d)*b+c^2+3.*c*d+d^2)*a+(d+c)*b^2+(c^2+3.*c*d+d^2)*b+c^2*d+c*d^2 = 0.], [a = .1666666667, b = .1666666667, c = .1666666667, d = .5000000000, lambda[1] = 0., -12.*((a^2+b^2+c^2+d^2)*a*b*c*d)^(1/2)+(b+c+d)*a^2+(b^2+(3.*c+3.*d)*b+c^2+3.*c*d+d^2)*a+(d+c)*b^2+(c^2+3.*c*d+d^2)*b+c^2*d+c*d^2 = 0.], [a = .5000000000, b = .1666666667, c = .1666666667, d = .1666666667, lambda[1] = 0., -12.*((a^2+b^2+c^2+d^2)*a*b*c*d)^(1/2)+(b+c+d)*a^2+(b^2+(3.*c+3.*d)*b+c^2+3.*c*d+d^2)*a+(d+c)*b^2+(c^2+3.*c*d+d^2)*b+c^2*d+c*d^2 = 0.], [a = .2118620934, b = .3644137199, c = .2118620934, d = .2118620934, lambda[1] = 0.2834790478e-2, -12.*((a^2+b^2+c^2+d^2)*a*b*c*d)^(1/2)+(b+c+d)*a^2+(b^2+(3.*c+3.*d)*b+c^2+3.*c*d+d^2)*a+(d+c)*b^2+(c^2+3.*c*d+d^2)*b+c^2*d+c*d^2 = 0.9449303017e-3], [a = 0.3692850681e-1, b = .8892144797, c = 0.3692850681e-1, d = 0.3692850681e-1, lambda[1] = 0.9303874297e-1, -12.*((a^2+b^2+c^2+d^2)*a*b*c*d)^(1/2)+(b+c+d)*a^2+(b^2+(3.*c+3.*d)*b+c^2+3.*c*d+d^2)*a+(d+c)*b^2+(c^2+3.*c*d+d^2)*b+c^2*d+c*d^2 = 0.3101291407e-1], [a = .8892144797, b = 0.3692850681e-1, c = 0.3692850681e-1, d = 0.3692850681e-1, lambda[1] = 0.9303874297e-1, -12.*((a^2+b^2+c^2+d^2)*a*b*c*d)^(1/2)+(b+c+d)*a^2+(b^2+(3.*c+3.*d)*b+c^2+3.*c*d+d^2)*a+(d+c)*b^2+(c^2+3.*c*d+d^2)*b+c^2*d+c*d^2 = 0.3101291407e-1], [a = .3644137199, b = .2118620934, c = .2118620934, d = .2118620934, lambda[1] = 0.2834790478e-2, -12.*((a^2+b^2+c^2+d^2)*a*b*c*d)^(1/2)+(b+c+d)*a^2+(b^2+(3.*c+3.*d)*b+c^2+3.*c*d+d^2)*a+(d+c)*b^2+(c^2+3.*c*d+d^2)*b+c^2*d+c*d^2 = 0.9449303017e-3], [a = 0.3692850681e-1, b = 0.3692850681e-1, c = 0.3692850681e-1, d = .8892144797, lambda[1] = 0.9303874297e-1, -12.*((a^2+b^2+c^2+d^2)*a*b*c*d)^(1/2)+(b+c+d)*a^2+(b^2+(3.*c+3.*d)*b+c^2+3.*c*d+d^2)*a+(d+c)*b^2+(c^2+3.*c*d+d^2)*b+c^2*d+c*d^2 = 0.3101291407e-1], [a = .2118620934, b = .2118620934, c = .2118620934, d = .3644137199, lambda[1] = 0.2834790478e-2, -12.*((a^2+b^2+c^2+d^2)*a*b*c*d)^(1/2)+(b+c+d)*a^2+(b^2+(3.*c+3.*d)*b+c^2+3.*c*d+d^2)*a+(d+c)*b^2+(c^2+3.*c*d+d^2)*b+c^2*d+c*d^2 = 0.9449303017e-3], [a = 0.3692850681e-1, b = 0.3692850681e-1, c = .8892144797, d = 0.3692850681e-1, lambda[1] = 0.9303874297e-1, -12.*((a^2+b^2+c^2+d^2)*a*b*c*d)^(1/2)+(b+c+d)*a^2+(b^2+(3.*c+3.*d)*b+c^2+3.*c*d+d^2)*a+(d+c)*b^2+(c^2+3.*c*d+d^2)*b+c^2*d+c*d^2 = 0.3101291407e-1], [a = .2118620934, b = .2118620934, c = .3644137199, d = .2118620934, lambda[1] = 0.2834790478e-2, -12.*((a^2+b^2+c^2+d^2)*a*b*c*d)^(1/2)+(b+c+d)*a^2+(b^2+(3.*c+3.*d)*b+c^2+3.*c*d+d^2)*a+(d+c)*b^2+(c^2+3.*c*d+d^2)*b+c^2*d+c*d^2 = 0.9449303017e-3]]](/view.aspx?sf=203207_post/cb85c4cc63d9e11f1ff3c355a143ef73.gif)
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(2) |
Indeed, the minimum value of the target function is exactly 0. Quod erat demonstrantum.
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