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\(\dfrac{ab}{6+2b+c}=\dfrac{ab}{a+b+c+2b+c}=\dfrac{ab}{\left(a+c\right)+\left(b+c\right)+2b}\le\dfrac{1}{9}\left(\dfrac{ab}{a+c}+\dfrac{ab}{b+c}+\dfrac{ab}{2b}\right)\)
Tương tự:
\(\dfrac{bc}{6+2c+a}\le\dfrac{1}{9}\left(\dfrac{bc}{a+b}+\dfrac{bc}{a+c}+\dfrac{bc}{2c}\right)\)
\(\dfrac{ac}{6+2a+b}\le\dfrac{1}{9}\left(\dfrac{ac}{a+b}+\dfrac{ac}{b+c}+\dfrac{ac}{2a}\right)\)
Cộng vế:
\(P\le\dfrac{1}{9}\left(\dfrac{ac+bc}{a+b}+\dfrac{ab+ac}{b+c}+\dfrac{ab+bc}{a+c}+\dfrac{a+b+c}{2}\right)=\dfrac{1}{6}\left(a+b+c\right)=1\)
TK: Cho các số thực dương a, b, c thỏa mãn a + b+ c = 3. Chứng minh rằng: \(\sqrt{2a^2+\frac{7}{b^2}}+\sqrt{2b^2+\frac{7}{... - Hoc24
\(\dfrac{bc}{a+b+c+a}\le\dfrac{bc}{4}\cdot\left(\dfrac{1}{a+b}+\dfrac{1}{a+c}\right)\\ \dfrac{ac}{b+c+a+b}\le\dfrac{ac}{4}\cdot\left(\dfrac{1}{b+c}+\dfrac{1}{a+b}\right)\\ \dfrac{ab}{a+c+b+c}\le\dfrac{ab}{4}\cdot\left(\dfrac{1}{a+c}+\dfrac{1}{b+c}\right)\\ \Leftrightarrow VT\le\dfrac{1}{a+b}\left(\dfrac{bc}{4}+\dfrac{ac}{4}\right)+\dfrac{1}{a+c}\left(\dfrac{bc}{4}+\dfrac{ab}{4}\right)+\dfrac{1}{b+c}\left(\dfrac{ac}{4}+\dfrac{ab}{4}\right)\\ =\dfrac{1}{a+b}\cdot\dfrac{c\left(a+b\right)}{4}+\dfrac{1}{a+c}\cdot\dfrac{b\left(a+c\right)}{4}+\dfrac{1}{b+c}\cdot\dfrac{a\left(b+c\right)}{4}\\ =\dfrac{c}{4}+\dfrac{b}{4}+\dfrac{a}{4}\\ =\dfrac{a+b+c}{4}\left(đfcm\right)\)
