Question

Chứng minh rằng:

a, \(a^2\)+\(b^2-2ab\ge0\)

b,\(\dfrac{a^2+b^2}{2}\ge ab\)

c,\(a\left(a+2\right)< \left(a+1\right)^2\)

d,\(m^2+n^2+2\ge2\left(m+n\right)\)

e, \(\left(a+b\right)\left(\dfrac{1}{a}+\dfrac{1}{b}\right)\ge4\)(với a > 0, b > 0)

Answer 1

a) Ta có: \(\left(a-b\right)^2\ge0\)

=>\(a^2+b^2-2ab\ge0\left(đpcm\right)\)

b) \(\left(a+b\right)^2\ge0\)

=> \(a^2+b^2+2ab\ge0\)

<=> \(a^2+b^2\ge-2ab\)

<=> \(\dfrac{a^2+b^2}{2}\ge ab\) (đpcm)

c) ta có: \(\left(a+1\right)^2=a^2+2a+1\)

\(a\left(a+2\right)=a^2+2a\)

Vậy từ 2 điều trên => \(a\left(a+2\right)< \left(a+1\right)^2\)

d) \(m^2+n^2+2\ge2\left(m+n\right)\) (*)

<=>m² - 2m +1 +n² - 2n +1 \(\ge0\)

<=> \(\left(m-1\right)^2+\left(n-1\right)^2\ge0\) (1)

(1) đúng => (*) đúng

d) Bạn ấy giải rồi ,mình không giải nữa

Answer 2

e) Theo BĐT cauchy ta có: \(\dfrac{a^2+b^2}{2}\ge ab\Rightarrow\dfrac{a^2+b^2}{ab}\ge2\)

\(\Leftrightarrow\dfrac{a}{b}+\dfrac{b}{a}\ge2\Leftrightarrow\left(\dfrac{a}{b}+1\right)+\left(\dfrac{b}{a}+1\right)\ge4\)

\(\Leftrightarrow\dfrac{a+b}{b}+\dfrac{a+b}{a}\ge4\)

\(\Rightarrow\left(a+b\right)\left(\dfrac{1}{b}+\dfrac{1}{a}\right)\ge4\) (đpcm)

Vậy..........