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Bất đẳng thức Bunhia [lớp 8]?

bài1.

cho x, y ,z>0 thoả mãn x^2 + y^2 + z^2 =3

Chứg mih rằg 1/(1+xy) + 1/(1+xz) + 1/(1+zy) >=3/2

bài 2.

cho a,b,c,d>0. CMR

1/a+1/b+4/c+16/d>=64/(a+b+c+d)

bài 2 chính là bđt Bun dạg:

a^2/x+b^2/y>=[(a+b)^2]/(x+y)

mìh địh áp dụg trực tiếp hok pýt koá đc hok nữa???????????

2 Answers

Rating
  • 9 years ago
    Favorite Answer

    bài 1

    áp dụng BĐT

    ta có : 1/(1+xy) +1 /( 1+xz)+ 1/( 1+yz) >= [( 1+1+1)^2] / 1+xy+1+yz+1+xz = 9 / 3+xy+yz+ xz

    ta ;ại có x^2 +y^2 +z^2 >= xy + yz +xz

    => 3>= xy+ yz +xz

    nên 3 + xy+ yz+xz =< 3+3 = 6

    9/ ( 3+xy+yz+ xz) >= 9/ 6 = 3/2

    bài 2 hướng đúng đấy bạn

    1/a+1/b+4/c+16/d >= ( 1+1+2+4)^2 / a+b+c+b = 8^2 / a+b+c+d = 64/(a+b+c+d)

  • 6 years ago

    Vậy dấu bằng xảy ra khi nào?

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