Question

Tìm x,y,z biết:
x + y + z - 6 = \(\sqrt{x-3}+\sqrt{y-4}+2\sqrt{z-5}\)

Answer 1

Lời giải:

PT \(\Leftrightarrow 2x+2y+2z-12=2\sqrt{x-3}+2\sqrt{y-4}+4\sqrt{z-5}\)

Áp dụng BĐT AM-GM ta có:

\(2\sqrt{x-3}=2\sqrt{\frac{1}{2}(2x-6)}\leq \frac{1}{2}+2x-6\)

\(2\sqrt{y-4}=2\sqrt{\frac{1}{2}(2y-8)}\leq \frac{1}{2}+2y-8\)

\(4\sqrt{z-5}=2\sqrt{2(2z-10)}\leq 2+2z-10\)

Cộng theo vế:

\(2\sqrt{x-3}+2\sqrt{y-4}+4\sqrt{z-5}\leq 2x+2y+2z-21\)

\(\Leftrightarrow 2x+2y+2z-12\leq 2x+2y+2z-21\)

\(\Leftrightarrow 9\leq 0\) (vô lý)

Do đó pt vô nghiệm.