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[19kvh97]

Tìm $m$ để pt sau có nghiệm

$x\sqrt{x}+\sqrt{x+12}=m(\sqrt{5-x}+\sqrt{4-x})$

[dang123]

Tìm $m$ để pt sau có nghiệm

$x\sqrt{x}+\sqrt{x+12}=m(\sqrt{5-x}+\sqrt{4-x})$

ĐKXĐ:     $0\leq x\leq 4$

PT $\Leftrightarrow \frac{x\sqrt{x}+\sqrt{x+12}}{\sqrt{5-x}+\sqrt{4-x}}=m$

Gọi f(x)=$\frac{x\sqrt{x}+\sqrt{x+12}}{\sqrt{5-x}+\sqrt{4-x}}$

f(x)'=$\frac{(\frac{3}{2}\sqrt{x}+\frac{1}{2\sqrt{x+12}})(\sqrt{5-x}+\sqrt{4-x})+(x\sqrt{x}+\sqrt{x+12})(\frac{1}{2\sqrt{5-x}}+\frac{1}{2\sqrt{4-x}})}{(\sqrt{5-x}+\sqrt{4-x})}$ >0 Với mọi   0=< x<=4

$\Rightarrow f(0)\leq f(x)\leq f(4)$

$2\sqrt{15}-4\sqrt{3} \leq m\leq 12$