作业帮若XYZ均为正整数,则(xy+yz)/[(x^2)+(y^2)+(z^2)]的最大值为
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![若XYZ均为正整数,则(xy+yz)/[(x^2)+(y^2)+(z^2)]的最大值为](/uploads/image/z/8599795-43-5.jpg?t=%E8%8B%A5XYZ%E5%9D%87%E4%B8%BA%E6%AD%A3%E6%95%B4%E6%95%B0%2C%E5%88%99%28xy%2Byz%29%2F%5B%28x%5E2%29%2B%28y%5E2%29%2B%28z%5E2%29%5D%E7%9A%84%E6%9C%80%E5%A4%A7%E5%80%BC%E4%B8%BA)
若XYZ均为正整数,则(xy+yz)/[(x^2)+(y^2)+(z^2)]的最大值为
若XYZ均为正整数,则(xy+yz)/[(x^2)+(y^2)+(z^2)]的最大值为
若XYZ均为正整数,则(xy+yz)/[(x^2)+(y^2)+(z^2)]的最大值为
(x^2)+(y^2)+(z^2)
= x^2 + 1/2y^2 + 1/2y^2 + z^2
≥ 2√(1/2)xy + 2√(1/2)yz
=√2 (xy+yz)
所以(xy+yz)/[(x^2)+(y^2)+(z^2)] ≤ √2/2
最大值为√2/2
当√2x = y = √2z时取得
(注：这里x,y,z应该是正数,而不是正整数,否则无法取得最大值√2/2,但可以无限接近)
(xy+yz)/[(x^2)+(y^2)+(z^2)]
=2(xy+yz)/[2(x^2)+2(y^2)+2(z^2)]
=2(xy+yz)/[(2(x^2)+(y^2))+((y^2)+2(z^2))]
2(x^2)+(y^2) >= 2(2(x^2)*(y^2))^(1/2) = 2xy* 2^(1/2) 当且仅当2x^2=y^2
(y^2)+2(z^2) >=...
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(xy+yz)/[(x^2)+(y^2)+(z^2)]
=2(xy+yz)/[2(x^2)+2(y^2)+2(z^2)]
=2(xy+yz)/[(2(x^2)+(y^2))+((y^2)+2(z^2))]
2(x^2)+(y^2) >= 2(2(x^2)*(y^2))^(1/2) = 2xy* 2^(1/2) 当且仅当2x^2=y^2
(y^2)+2(z^2) >= 2(2(z^2)*(y^2))^(1/2) = 2yz* 2^(1/2)当且仅当2z^2=y^2
2(x^2)+(y^2) + (y^2)+2(z^2) >= 2*2^(1/2) *(xy + yz) 此时要求以上两个条件都成立
2(xy+yz)/[(2(x^2)+(y^2))+((y^2)+2(z^2))] <= 2(xy+yz)/(2*2^(1/2) *(xy + yz)) = 2^(-1/2)
即最大值为根号2 分之一
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