Question

Cho \(P=\dfrac{x}{\sqrt{x}-5}\)

Khi x > 25. Tìm GTNN của P

Answer 1

\(P=\dfrac{x}{\sqrt{x}-5}=\dfrac{x-25}{\sqrt{x}-5}+\dfrac{25}{\sqrt{x}-5}=\sqrt{x}+5+\dfrac{25}{\sqrt{x}-5}=\left(\sqrt{x}-5+\dfrac{25}{\sqrt{x}-5}\right)+10\)

Áp dụng BĐT AM-GM có;
\(\sqrt{x}-5+\dfrac{25}{\sqrt{x}-5}\ge10\)

\(\Rightarrow P\ge10+10=20\)

Dấu "=" xảy ra khi\(\sqrt{x}-5=\dfrac{25}{\sqrt{x}-5}\)\(\Leftrightarrow x=100\)(tm)

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