Question

Tìm GTNN của C = \(\frac{x+8}{\sqrt{x}+1}\)đkxđ \(x\ge0;x\ne9\)

Answer 1

\(C=\frac{x-1+9}{\sqrt{x}+1}=\sqrt{x}-1+\frac{9}{\sqrt{x}+1}=\sqrt{x}+1+\frac{9}{\sqrt{x}+1}-2\)

Áp dụng BĐT Cauchy:

\(C\ge2\sqrt{\frac{\left(\sqrt{x}+1\right).9}{\sqrt{x}+1}}-2=4\)

\(\Rightarrow C_{min}=4\) khi \(\left(\sqrt{x}+1\right)^2=9\Rightarrow x=4\)