Question

[Nguyễn Anh Thư]

Tìm GTNN của các biểu thức sau: 
f) F= \(\frac{x+15}{\sqrt{x}+1}\)
g) G= \(\frac{x-5}{\sqrt{x}+3}\)

Answer 1

[Nguyễn Minh Quang]

\(F=\frac{x-1+16}{\sqrt{x}+1}=\sqrt{x}-1+\frac{16}{\sqrt{x}+1}=\sqrt{x}+1+\frac{16}{\sqrt{x}+1}-2\)

\(\ge2\sqrt{\left(\left(\sqrt{x}+1\right).\frac{16}{\sqrt{x}+1}\right)}-2=8-2=6\) vậy GTNN của F=6 khi \(\sqrt{x}+1=\frac{16}{\sqrt{x}+1}\Leftrightarrow x=9\)

​\(G=\frac{x-9+4}{\sqrt{x}+3}=\sqrt{x}-3+\frac{4}{\sqrt{x}+3}=\sqrt{x}+3+\frac{4}{\sqrt{x}+3}-6=\frac{5}{9}\left(\sqrt{x}+3\right)+\frac{4}{9}\left(\sqrt{x}+3\right)+\frac{4}{\sqrt{x}+3}-6\)​

\(\ge\frac{5}{9}\left(\sqrt{x}+3\right)+2\sqrt{\left(\frac{4}{9}\left(\sqrt{x}+3\right).\frac{4}{\sqrt{x}+3}\right)}-6\ge\frac{5}{3}+\frac{8}{3}-6=-\frac{5}{3}\) vậy GTNN G =- 5/3 khi x=0

Answer 2

[l҉o҉n҉g҉ d҉z҉]

\(F=\frac{x-1+16}{\sqrt{x}+1}=\frac{x-1}{\sqrt{x}+1}+\frac{16}{\sqrt{x}+1}=\sqrt{x}-1+\frac{16}{\sqrt{x}+1}\)

\(=\left[\left(\sqrt{x}+1\right)+\frac{16}{\sqrt{x}+1}\right]-2\ge2\sqrt{\left(\sqrt{x}+1\right)\cdot\frac{16}{\sqrt{x}+1}}-2=6\)

Dấu "=" xảy ra <=> x = 9