Lời giải:
Áp dụng BĐT Cô-si cho các số không âm:
\(a^2+2=(a^2+1)+1\geq 2\sqrt{(a^2+1).1}=2\sqrt{a^2+1}\)
\(\Rightarrow \frac{a^2+2}{\sqrt{a^2+1}}\geq \frac{2\sqrt{a^2+1}}{\sqrt{a^2+1}}=2\) (đpcm)
Dấu "=" xảy ra khi \(a^2+1=1\Leftrightarrow a=0\)
CMR, ∀n ≥ 1, n ∈ N : \(\dfrac{1}{2}\)+\(\dfrac{1}{3\sqrt{2}}\)+\(\dfrac{1}{4\sqrt{3}}\)+....+ \(\dfrac{1}{\left(n+1\right)\sqrt{n}}\)<2
(4)Bài 1:Với \(\forall\) a>b>0. CMR: a+ \(\frac{1}{b\left(a-b\right)}\ge3\)
(7) Bài 2: Cho a,b,c \(\ne\) 0 .CMR: \(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\)
(8) Bài 3: Cho a,b,c>0 thõa mãn abc=1
CMR: \(\frac{b+c}{\sqrt{a}}+\frac{c+a}{\sqrt{b}}+\frac{a+b}{\sqrt{c}}\ge\sqrt{a}+\sqrt{b}+\sqrt{c}+3\)
Cho các số dương a, b, c thỏa mãn: \(\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2}=\sqrt{2011}\). CMR: \(\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}++\dfrac{c^2}{a+b}\ge\dfrac{1}{2}\sqrt{\dfrac{2011}{2}}\)
Cho các số dương a, b, c thỏa mãn: \(\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2}=\sqrt{2011}\). CMR: \(\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}\ge\dfrac{1}{2}\sqrt{\dfrac{2011}{2}}\)
cho a,b,c dương thỏa mãn \(\sqrt{a^2+b^2}+\sqrt{b^2+c^2}+\sqrt{c^2+a^2}=\sqrt{2011}\).
CMR: \(\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}\ge\dfrac{\sqrt{2011}}{2}\)
Cho a, b, c>0 và a+b+c\(\ge3\)
Cmr:
\(\dfrac{a^2}{a+\sqrt{bc}}+\dfrac{b^2}{b+\sqrt{ac}}+\dfrac{c^2}{c+\sqrt{ab}}\ge\dfrac{3}{2}\)
(2) Bài 1: Với \(\forall\) a>1.CMR: \(a+\frac{1}{a-1}\ge3\)
(3)Bài 2:Với \(\forall\) a,b >0 .CMR: \(\frac{a^2+2}{\sqrt{a^2+1}}\ge2\)
(5) Bài 3: Với \(\forall\) a>b>0. CMR: \(a+\frac{4}{\left(a+b\right)\left(b+1\right)^2}\ge3\)
a)Cho 0 < c ; c < b ; b < a . CMR:\(\sqrt{c\left(a-c\right)}+\sqrt{b\left(b-c\right)}\le\sqrt{ab}\)
b)Cho \(x\ge1;y\ge1\). CMR:\(\dfrac{1}{1+x^2}+\dfrac{1}{1+y^2}\ge\dfrac{2}{1+xy}\)
Cho a,b,c là các số dương. CMR
\(\dfrac{a}{\sqrt{2b^2+2c^2-a^2}}+\dfrac{b}{\sqrt{2c^2+2a^2-b^2}}+\dfrac{c}{\sqrt{2a^2+2b^2-c^2}}\ge\sqrt{3}\)
