数学题:难题设2006x3=2007y3=2008z3,且xyz>0,3√2006x2+2007y2+2008z2 =3√2006+3√2007+ 3√2008,求1/x+1/y+1/z
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![数学题:难题设2006x3=2007y3=2008z3,且xyz>0,3√2006x2+2007y2+2008z2 =3√2006+3√2007+ 3√2008,求1/x+1/y+1/z](/uploads/image/z/5429624-32-4.jpg?t=%E6%95%B0%E5%AD%A6%E9%A2%98%EF%BC%9A%E9%9A%BE%E9%A2%98%E8%AE%BE2006x3%3D2007y3%3D2008z3%2C%E4%B8%94xyz%EF%BC%9E0%2C3%E2%88%9A2006x2%2B2007y2%2B2008z2++%3D3%E2%88%9A2006%2B3%E2%88%9A2007%2B++++3%E2%88%9A2008%2C%E6%B1%821%EF%BC%8Fx%2B1%EF%BC%8Fy%2B1%EF%BC%8Fz)
数学题:难题设2006x3=2007y3=2008z3,且xyz>0,3√2006x2+2007y2+2008z2 =3√2006+3√2007+ 3√2008,求1/x+1/y+1/z
数学题:难题
设2006x3=2007y3=2008z3,且xyz>0,3√2006x2+2007y2+2008z2 =3√2006+3√2007+ 3√2008,求1/x+1/y+1/z
数学题:难题设2006x3=2007y3=2008z3,且xyz>0,3√2006x2+2007y2+2008z2 =3√2006+3√2007+ 3√2008,求1/x+1/y+1/z
设2006x^3=2007y^3=2008z^3=k
(2006x2+2007y2+2008z2)^(1/3)=2006^(1/3)+2007^(1/3)+2008^(1/3)
==>
(k/x+k/y+k/z)^(1/3)=(k/x^3)^(1/3)+(k/y^3)^(1/3)+(k/z^3)^(1/3)
==>k^(1/3)(1/x+1/y+1/z)^(1/3)=k^(1/3)(1/x+1/y+1/z)
==>1/x+1/y+1/z=(1/x+1/y+1/z)^(1/3)
==>1/x+1/y+1/z=1,或0(xyz>0,所以舍去),或-1(xyz>0,所以舍去)
故1/x+1/y+1/z=1
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