Question

Tìm GTNN của biểu thức sau:
A= a+1/a, a>=3

Answer 1

\(A=\dfrac{a+1}{a}\) hay \(A=a+\dfrac{1}{a}\)?

Answer 2

Áp dụng BĐT cosi cho \(a>0\left(a\ge3\right)\)

\(A=a+\dfrac{1}{a}=\dfrac{a}{9}+\dfrac{8a}{9}+\dfrac{1}{a}\ge2\sqrt{\dfrac{a}{9}\cdot\dfrac{1}{a}}+\dfrac{8a}{9}=2\cdot\dfrac{1}{3}+\dfrac{8a}{9}\ge\dfrac{2}{3}+\dfrac{8\cdot3}{9}=\dfrac{10}{3}\)

Dấu \("="\Leftrightarrow a=3\)