设a>b>c>0,求2a^2+1/ab+1/a(a-b)-10ac+25c^2的最小值

设a>b>c>0,求2a^2+1/ab+1/a(a-b)-10ac+25c^2的最小值
设a>b>c>0,求2a^2+1/ab+1/a(a-b)-10ac+25c^2的最小值

设a>b>c>0,求2a^2+1/ab+1/a(a-b)-10ac+25c^2的最小值
∵b(a-b)≤(a/2)^2=a^2/4可得
2a^2+1/ab+1/a(a-b)-10ac+25c^2
=a^2+1/ab+1/a(a-b)+a^2-10ac+25c^2
=a^2+1/b(a-b)+a^2-10ac+25c^2
≥a^2+4/a^2+(a-5c)^2
≥4
当且仅当b=a-b,a=5c,a^2=2等号成立