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已知x>1,y>0,且2/(x%1)+1/y =2,求x+y最小值

其实题目是关系到“1”的妙用。 对于1/x+9/y=2,两边除以2,则 (1/2)·(1/x+9/y)=1, 所以, x+y=1·(x+y) =(1/2)(1/x+9/y)·(x+y) =(1/2)(10+y/x+9x/y) ≥(1/2)[10+2√(y/x·9x/y)] =(1/2)(10+6) =8, 即1/x+9/y=2且y/x=9x/y, x=2,y=6时, 所求x+y最...

x>0 y>0 则x+y>0 xy>0 1/x+1/y=2 (x+y)/(xy)=2 x+y=2xy 由均值不等式得4xy≤(x+y)²,xy≤(x+y)²/4 因此x+y≤2(x+y)²/4 整理,得 (x+y)²-2(x+y)≥0 (x+y)(x+y-2)≥0 x+y≥2或x+y≤0(舍去) x+y的最小值是2,此时x=y=1。

x+y=1 x^2+y^2+2xy=1 x^2+y^2=1-2xy 0

x>y>0,x+y≤2, ∴2/(x+3y)+1/(x-y) =(√2)²/(x+3y)+1²/(x-y) ≥(√2+1)²/[(x+3y)+(x-y)] =(3+2√2)/[2(x+y)] ≥(3+2√2)/4. 故(x+3y):√2=(x-y):1且x+y=2, 即x=-1+2√2,y=3-2√2时, 所求最小值为: (3+2√2)/4。

由x>0,y>0, 且2/x+1/y=1 x+2y=(x+2y)(2/x+1/y) =2+x/y+4y/x+2 =4+x/y+4y/x ≥4+4 即当x/y=4y/x, x=2y时, 2/x+1/y=1 2/2y+1/y=1 2/y=1,∴y=2,x=4 x+2y有最小值8

x>y>0且x+y≤2, 故依均值不等式得 2/(x+3y)+1/(x-y) =1·[2/(x+3y)+1/(x-y)] =[(x+3y)+(x-y)]/4·[2/(x+3y)+1/(x-y)] =3/4+(1/4)·[(x+3y)/(x-y)+2(x-y)/(x+3y)] ≥3/4+(1/4)·2√[(x+3y)/(x-y)·2(x-y)/(x+3y)] =(3+2√2)/4. ∴(x+3y)/(x-y)=2(x-y)/(x+3...

∵x+y=1 ∴(x+2)+(y+1)=4 x^2/(x+2)+y^2/(y+1) =[(x+2)-2]^2/(x+2)+[(y+1)-1]^2/(y+1) =[(x+2)^2-4(x+2)+4]/(x+2)+[(y+1)^2-2(y+1)+1]/(y+1) =(x+2)-4+4/(x+2)+(y+1)-2+1/(y+1) =4/(x+2)+1/(y+1)+(x+y-3) =[4/(x+2)+1/(y+1)] *[(x+2)+(y+1)]/4 -2 ...

y=(1/x+1/y)(x+2y)=1+2+2y/x+x/y=3+(2y/x+x/y) ≥ 3+2√(2y/x)*(x/y)=3+2√2 当且仅当,2y/x=x/y,x=√2y时,等号成立 最小值=3+2√2 y=1/x+1/y的最小值=3+2√2

1/x2+1/y2+1/xy≥3 * 3^√ ̄1/x2*y2*xy =3 * 3^√ ̄1/x^3*y^3 x+y=2≥2√ ̄xy 所以√ ̄xy ≤1 所以 3 * 3^√ ̄1/x^3*y^3≤3

2/x+8/y =2(x+y)/x+8(x+y)/y =2+2y/x+8x/y+8 =10+2y/x+8x/y ≥10+2√2y/x·8x/y =10+8 =18 此时 2y/x=8x/y y平方=4x平方 y=2x 3x=1 x=1/3,y=2/3 最小值=18.

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