设a.b.c.均为正数,且a+b+c=1求证1/a+1/b+1/c大于等于9
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设a.b.c.均为正数,且a+b+c=1求证1/a+1/b+1/c大于等于9
![设a.b.c.均为正数,且a+b+c=1求证1/a+1/b+1/c大于等于9](/uploads/image/z/16517389-13-9.jpg?t=%E8%AE%BEa.b.c.%E5%9D%87%E4%B8%BA%E6%AD%A3%E6%95%B0%2C%E4%B8%94a%2Bb%2Bc%3D1%E6%B1%82%E8%AF%811%2Fa%2B1%2Fb%2B1%2Fc%E5%A4%A7%E4%BA%8E%E7%AD%89%E4%BA%8E9)
1/a+1/b+1/c
=(a+b+c)/a+(a+b+c)/b+(a+b+c)/c
=1+(b+c)/a+1(a+c)/b+1(a+b)/c
=3+b/c+c/b+a/c+c/a+a/b+b/a
>=3+2+2+2
=9
取等号时a=b=c=1/3
=(a+b+c)/a+(a+b+c)/b+(a+b+c)/c
=1+(b+c)/a+1(a+c)/b+1(a+b)/c
=3+b/c+c/b+a/c+c/a+a/b+b/a
>=3+2+2+2
=9
取等号时a=b=c=1/3
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