b) Cho a, b biết : a+b=1. Chứng minh 1/a+1 + 1/b+1 bé hơn hoặc bằng 4/3 31/07/2021 Bởi Sarah b) Cho a, b biết : a+b=1. Chứng minh 1/a+1 + 1/b+1 bé hơn hoặc bằng 4/3
Áp dụng BĐT dạng \(\dfrac{1}{a}+\dfrac{1}{b}\ge \dfrac{4}{a+b}\) \(→\dfrac{1}{a+1}+\dfrac{1}{b+1}\ge \dfrac{4}{a+b+1+1}\) mà \(a+b=1\) \(→\dfrac{1}{a+1}+\dfrac{1}{b+1}\ge \dfrac{4}{1+1+1}=\dfrac{4}{3}\) Bình luận
$\frac{1}{a}$ +1+$\frac{1}{b}$ +1$\leq$ $\frac{4}{3}$\\<=>$\frac{a+b}{ab}$ +2$\leq$ $\frac{4}{3}$ \\mà a+b=1 \\=>$\frac{1}{ab}$ +2$\leq$ $\frac{4}{3}$ \\ $\frac{1}{ab}$ $\leq$ $\frac{-2}{3}$ \\ab=$\frac{-3}{2}$\\ vậy $\frac{a+b}{ab}$ +2$\leq$ $\frac{4}{3}$ <=> 1: $\frac{-3}{2}$+2 $\leq$ $\frac{4}{3}$ <=>$\frac{4}{3}$ $\leq$ $\frac{4}{3}$ ™ Bình luận
Áp dụng BĐT dạng \(\dfrac{1}{a}+\dfrac{1}{b}\ge \dfrac{4}{a+b}\)
\(→\dfrac{1}{a+1}+\dfrac{1}{b+1}\ge \dfrac{4}{a+b+1+1}\)
mà \(a+b=1\)
\(→\dfrac{1}{a+1}+\dfrac{1}{b+1}\ge \dfrac{4}{1+1+1}=\dfrac{4}{3}\)
$\frac{1}{a}$ +1+$\frac{1}{b}$ +1$\leq$ $\frac{4}{3}$
\\<=>$\frac{a+b}{ab}$ +2$\leq$ $\frac{4}{3}$
\\mà a+b=1
\\=>$\frac{1}{ab}$ +2$\leq$ $\frac{4}{3}$
\\ $\frac{1}{ab}$ $\leq$ $\frac{-2}{3}$ \\ab=$\frac{-3}{2}$
\\ vậy $\frac{a+b}{ab}$ +2$\leq$ $\frac{4}{3}$
<=> 1: $\frac{-3}{2}$+2 $\leq$ $\frac{4}{3}$
<=>$\frac{4}{3}$ $\leq$ $\frac{4}{3}$ ™