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\(\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\ge0\) ; \(\forall a;b;c\)
\(\Leftrightarrow a^2+b^2+c^2\ge ab+bc+ca\)
\(\Rightarrow ab+bc+ca\le1\)
\(\Rightarrow P_{max}=1\) khi \(a=b=c\)
Lại có:
\(\left(a+b+c\right)^2\ge0\) ; \(\forall a;b;c\)
\(\Leftrightarrow a^2+b^2+c^2+2\left(ab+bc+ca\right)\ge0\)
\(\Leftrightarrow ab+bc+ca\ge-\dfrac{a^2+b^2+c^2}{2}=-\dfrac{1}{2}\)
\(P_{min}=-\dfrac{1}{2}\) khi \(a+b+c=0\)
Ta có:
\(\left(a-b\right)^2\ge0\Leftrightarrow a^2-2ab+b^2\ge0\Leftrightarrow a^2+b^2\ge2ab\)
Thiết lập 2 BĐT tương tự ta có:
\(b^2+c^2\ge2bc;c^2+a^2\ge2ca\)
Và \(\left(a-1\right)^2\ge0\Leftrightarrow a^2-2a+1\Leftrightarrow a^2+1\ge2a\)
Và tương tự \(b^2+1\ge2b;c^2+1\ge2c\)
Cộng theo vế các BĐT trên ta có:
\(2ab+2bc+2ca+2a+2b+2c\le3a^2+3b^2+3c^2+3\)
\(\Leftrightarrow2\left(ab+bc+ca+a+b+c\right)\le3\left(a^2+b^2+c^2+1\right)\)
\(\Leftrightarrow2\left(ab+bc+ca+a+b+c\right)\le12\)
\(\Leftrightarrow ab+bc+ca+a+b+c\le6\)
Đẳng thức xảy ra khi \(a=b=c=1\)
Khi đó \(A=\dfrac{a^{30}+b^4+c^{1975}}{a^{30}+b^4+c^{2014}}=\dfrac{1+1+1}{1+1+1}=1\)
\(9=3a^2+2b^2+2bc+2c^2=\left(a+b+c\right)^2+2a^2+b^2+c^2-2a\left(b+c\right)\)
\(\Rightarrow9\ge\left(a+b+c\right)^2+2a^2+\dfrac{1}{2}\left(b+c\right)^2-2a\left(b+c\right)\)
\(\Rightarrow9\ge\left(a+b+c\right)^2+\dfrac{1}{2}\left(2a-b-c\right)^2\ge\left(a+b+c\right)^2\)
\(\Rightarrow-3\le a+b+c\le3\)
\(T_{max}=3\) khi \(a=b=c=1\)
\(T_{min}=-3\) khi \(a=b=c=-1\)
Đặt \(P=a^2+b^2+c^2+ab+bc+ca\)
\(P=\dfrac{1}{2}\left(a+b+c\right)^2+\dfrac{1}{2}\left(a^2+b^2+c^2\right)\)
\(P\ge\dfrac{1}{2}\left(a+b+c\right)^2+\dfrac{1}{6}\left(a+b+c\right)^2=6\)
Dấu "=" xảy ra khi \(a=b=c=1\)
Từ giả thiết:
\(a^2=2\left(b^2+c^2\right)\ge\left(b+c\right)^2\Rightarrow\left(\dfrac{a}{b+c}\right)^2\ge1\Rightarrow\dfrac{a}{b+c}\ge1\)
\(P=\dfrac{a}{b+c}+\dfrac{b^2}{bc+ab}+\dfrac{c^2}{ac+bc}\ge\dfrac{a}{b+c}+\dfrac{\left(b+c\right)^2}{a\left(b+c\right)+2bc}\ge\dfrac{a}{b+c}+\dfrac{\left(b+c\right)^2}{a\left(b+c\right)+\dfrac{1}{2}\left(b+c\right)^2}\)
\(P\ge\dfrac{a}{b+c}+\dfrac{1}{\dfrac{a}{b+c}+\dfrac{1}{2}}\)
Đặt \(\dfrac{a}{b+c}=x\ge1\)
\(\Rightarrow P\ge x+\dfrac{1}{x+\dfrac{1}{2}}=\dfrac{4}{9}\left(x+\dfrac{1}{2}\right)+\dfrac{1}{x+\dfrac{1}{2}}+\dfrac{5}{9}x-\dfrac{2}{9}\)
\(P\ge2\sqrt{\dfrac{4}{9}\left(x+\dfrac{1}{2}\right).\dfrac{1}{\left(x+\dfrac{1}{2}\right)}}+\dfrac{5}{9}.1-\dfrac{2}{9}=\dfrac{5}{3}\)
\(P_{min}=\dfrac{5}{3}\) khi \(x=1\) hay \(a=2b=2c\)
Ta có:
\(\left(a-b\right)^2+\left(a-c\right)^2+\left(b-c\right)^2\ge0\)
\(\Rightarrow2\left(a^2+b^2+c^2\right)-2\left(ab+ac+bc\right)\ge0\)
\(\Rightarrow ab+ac+bc\le\dfrac{2.3}{2}=3\) (1)
Lại có: \(a^2+1+b^2+1+c^2+1\ge2a+2b+2c\)
\(\Rightarrow a+b+c\le\dfrac{a^2+b^2+c^2+3}{2}=3\) (2)
Cộng vế với vế của (1) và (2) ta được:
\(a+b+c+ab+ac+bc\le6\)
Dấu "=" xảy ra khi và chỉ khi \(a=b=c=1\)
\(\Rightarrow A=\dfrac{1^{30}+1^4+1^{1975}}{1^{30}+1^4+1^{2017}}=\dfrac{3}{3}=1\)