(1/2)若a.b.c是互不相等的正数,求证:a^ 4+b^ a+c ^ 4> a ^ 2b^ 2 +b^ 2c^ 2

a^4+b^4+c^4>a^2b^2+b^2c^2+c^2a^2>abc(a+b+c)根据(a-b)^2≥0(a=b时=0)a≠b(a-b)^2>0a^2-2ab+b^2>0a^2+b^2>2aba^4+b^4>2a^2b^2……①b^4+c^4>2b^2c^2……②c^4+a^4>2c^2a^2……③①+②+③2a^4+2b^4+2c^4>2a^2b^2+2b^2c^2+2c^2a^2a^4+b^4+c^4>a^2b^2+b^2c^2+c^2a^2 a^2b^2+b^2c^2>2abbc……④b^2c^2+c^2a^2>2bcca……⑤c^2a^2+a^2b^2>2caab……⑥④+⑤+⑥2a^2b^2+2b^2c^2+2c^2a^2>2abbc+2bcca+2caaba^2b^2+b^2c^2+c^2a^2>abc(a+b+c)a^4+b^4+c^4>a^2b^2+b^2c^2+c^2a^2>abc(a+b+c)