Cho $x,y,z\in [0;2]$. CMR
$2(x+y+z)\leq xy+yz+zx+4$
Lời giải Duc3290, 01-12-2023 - 23:06
Cho $x,y,z\in [0;2]$. CMR
$2(x+y+z)\leq xy+yz+zx+4$
Do $x,y,z\in[0;2] $ nên
$$(x-2)(y-2)(z-2)\leq 0$$
$$\leftrightarrow xyz +4(x+y+z)-2(xy+yz+zx)-8\leq 0$$
$$\leftrightarrow xyz +4(x+y+z)\leq2(xy+yz+zx+4)$$
Vì $xyz\geq 0 \rightarrow 4(x+y+z)\leq2(xy+yz+zx+4) \leftrightarrow 2(x+y+z)\leq xy+yz+zx +4$
Đi đến bài viết »Cho $x,y,z\in [0;2]$. CMR
$2(x+y+z)\leq xy+yz+zx+4$
Cho $x,y,z\in [0;2]$. CMR
$2(x+y+z)\leq xy+yz+zx+4$
Do $x,y,z\in[0;2] $ nên
$$(x-2)(y-2)(z-2)\leq 0$$
$$\leftrightarrow xyz +4(x+y+z)-2(xy+yz+zx)-8\leq 0$$
$$\leftrightarrow xyz +4(x+y+z)\leq2(xy+yz+zx+4)$$
Vì $xyz\geq 0 \rightarrow 4(x+y+z)\leq2(xy+yz+zx+4) \leftrightarrow 2(x+y+z)\leq xy+yz+zx +4$
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