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a/ \(\left(a^2+b^2\right)+\left(a^2+1\right)+\left(b^2+1\right)\ge2ab+2a+2b\)
\(\Leftrightarrow a^2+b^2+1\ge ab+a+b\)
b/ \(\dfrac{1}{a}+\dfrac{1}{b}\ge\dfrac{4}{a+b}\)
\(\Leftrightarrow a^2-2ab+b^2\ge0\)
\(\Leftrightarrow\left(a-b\right)^2\ge0\) đúng
c/ \(M=x^4-6x^3+13x^2-12x-5\)
Đặt \(x^2-3x=a\)thì ta có:
\(M=a^2+4a-5=\left(a+2\right)^2-9\ge-9\)
Dấu = xảy ra khi:
\(x^2-3x+2=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=1\\x=2\end{matrix}\right.\)
5. phân tích ra : \(1+\dfrac{a}{b}+\dfrac{b}{a}+1\)
áp dụng bđ cosy
\(\dfrac{a}{b}+\dfrac{b}{a}\ge2\sqrt{\dfrac{a}{b}.\dfrac{b}{a}}=2\)
=> đpcm
6. \(x^2-x+1=x^2-2.\dfrac{1}{2}.x+\dfrac{1}{4}+\dfrac{3}{4}=\left(x-\dfrac{1}{2}\right)^2+\dfrac{3}{4}>0\)
hay với mọi x thuộc R đều là nghiệm của bpt
7.áp dụng bđt cosy
\(a^4+b^4+c^4+d^4\ge2\sqrt{a^2.b^2.c^2.d^2}=4abcd\left(đpcm\right)\)
Ta có:
\(\dfrac{bc}{a}+\dfrac{ac}{b}=c\left(\dfrac{b}{a}+\dfrac{a}{b}\right)\ge2c\)
Chứng minh tương tự, ta có:
\(\dfrac{bc}{a}+\dfrac{ab}{c}\ge2b\)
\(\dfrac{ac}{b}+\dfrac{ab}{c}\ge2a\)
\(\Rightarrow2\left(\dfrac{bc}{a}+\dfrac{ac}{b}+\dfrac{ab}{c}\right)\ge2\left(a+b+c\right)\)
\(\Rightarrow\dfrac{bc}{a}+\dfrac{ac}{b}+\dfrac{ab}{c}\ge a+b+c\)
Dấu = xảy ra khi a = b = c
Đầu tiên ta cm bđt:\(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\ge\dfrac{9}{a+b+c}\)(tự cm)
Áp dụng ta có:
\(A=\dfrac{1}{1+ab}+\dfrac{1}{1+bc}+\dfrac{1}{1+ca}\ge\dfrac{9}{3+ab+bc+ca}\)
Cần cm:\(ab+bc+ca\le3\)
Hay \(ab+bc+ca\le\dfrac{\left(a+b+c\right)^2}{3}\)
\(\Leftrightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\ge0\)(luôn đúng)
=>đpcm
Áp dụng BĐT Cô - Si , ta có :
\(\dfrac{a}{b^2}+\dfrac{1}{a}\) ≥ \(2\sqrt{\dfrac{a}{b^2}.\dfrac{1}{a}}=2.\dfrac{1}{b}\left(a,b>0\right)\left(1\right)\)
\(\dfrac{b}{c^2}+\dfrac{1}{b}\text{ ≥ }2\sqrt{\dfrac{b}{c^2}.\dfrac{1}{b}}=2.\dfrac{1}{c}\left(b,c>0\right)\left(2\right)\)
\(\dfrac{c}{a^2}+\dfrac{1}{c}\text{≥}2\sqrt{\dfrac{c}{a^2}.\dfrac{1}{c}}=2.\dfrac{1}{a}\left(a,c>0\right)\left(3\right)\)
Từ ( 1 ; 2 ; 3) Ta có :
\(\dfrac{a}{b^2}+\dfrac{b}{c^2}+\dfrac{c}{a^2}+\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\) ≥ \(2\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)\)
⇔\(\dfrac{a}{b^2}+\dfrac{b}{c^2}+\dfrac{c}{a^2}\) ≥ \(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\)
Áp dụng bất đẳng thức Cauchy-Schwarz:
\(A=\dfrac{1}{1+ab}+\dfrac{1}{1+bc}+\dfrac{1}{1+ca}\)
\(A\ge\dfrac{\left(1+1+1\right)^2}{3+ab+bc+ac}=\dfrac{9}{3+ab+bc+ac}\)
Mặt khác,theo hệ quả AM-GM: \(ab+bc+ac\le\dfrac{\left(a+b+c\right)^2}{3}\le\dfrac{3^2}{3}=3\)
\(\Rightarrow\dfrac{9}{3+ab+bc+ac}\ge\dfrac{9}{3+3}=\dfrac{9}{6}=\dfrac{3}{2}\)
Dấu "=" xảy ra khi: \(a=b=c=1\)
a)Áp dụng bđt Cô-si:
\(\dfrac{a}{b}+\dfrac{b}{a}-1+\dfrac{ab}{a^2-ab+b^2}=\dfrac{a^2+b^2-ab}{ab}+\dfrac{ab}{a^2-ab+b^2}\ge2\sqrt{\dfrac{a^2+b^2-ab}{ab}.\dfrac{ab}{a^2-ab+b^2}}=2\)
=>\(\dfrac{a}{b}+\dfrac{b}{a}+\dfrac{ab}{a^2-ab+b^2}\ge3\)
Dấu "=" xảy ra khi a=b=1
b) bđt sai rồi
Áp dụng BĐT Cô - Si vào bài toán , ta có :
\(\dfrac{1}{a}+\dfrac{1}{b}\) ≥ \(\dfrac{2}{\sqrt{ab}}\)
⇔\(\left(a+b\right)\left(\dfrac{1}{a}+\dfrac{1}{b}\right)\) ≥ \(\dfrac{2}{\sqrt{ab}}.2\sqrt{ab}\)
⇔\(\left(a+b\right)\left(\dfrac{1}{a}+\dfrac{1}{b}\right)\) ≥ 4
⇔ \(\dfrac{1}{a}+\dfrac{1}{b}\) ≥ \(\dfrac{4}{a+b}\)
Dấu " = " xảy ra khi : a = b
Áp dụng BĐT Cô - si dạng Engel , ta có :
\(\dfrac{1}{a}+\dfrac{1}{b}\) ≥ \(\dfrac{\left(1+1\right)^2}{a+b}=\dfrac{4}{a+b}\)
Dấu " = " xảy ra khi : a = b