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\(4=2a^2+\dfrac{1}{a^2}+\dfrac{b^2}{4}=\left(a^2+\dfrac{1}{a^2}-2\right)+\left(a^2+\dfrac{b^2}{4}+ab\right)-ab+2\)
\(\Rightarrow4=\left(a-\dfrac{1}{a}\right)^2+\left(a+\dfrac{b}{2}\right)^2-ab+2\)
\(\Rightarrow ab=\left(a-\dfrac{1}{a}\right)^2+\left(a+\dfrac{b}{2}\right)^2-2\ge-2\)
\(M_{min}=-2\) khi \(\left\{{}\begin{matrix}a-\dfrac{1}{a}=0\\a+\dfrac{b}{2}=0\end{matrix}\right.\) \(\Rightarrow\left(a;b\right)=\left(1;-2\right);\left(-1;2\right)\)
Hi vọng là tìm GTLN:
Không mất tính tổng quát, giả sử b, c cùng phía với 1 \(\Rightarrow\left(b-1\right)\left(c-1\right)\ge0\Leftrightarrow bc\ge b+c-1\).
Áp dụng bất đẳng thức AM - GM ta có:
\(4=a^2+b^2+c^2+abc\ge a^2+2bc+abc\Leftrightarrow2bc+abc\le4-a^2\Leftrightarrow bc\left(a+2\right)\le\left(2-a\right)\left(a+2\right)\Leftrightarrow bc+a\le2\)
\(\Rightarrow a+b+c\le3\).
Áp dụng bất đẳng thức Schwarz ta có:
\(P\le\dfrac{ab}{9}\left(\dfrac{1}{a}+\dfrac{2}{b}\right)+\dfrac{bc}{9}\left(\dfrac{1}{b}+\dfrac{2}{c}\right)+\dfrac{ca}{9}\left(\dfrac{1}{c}+\dfrac{2}{a}\right)=\dfrac{1}{9}.3\left(a+b+c\right)=\dfrac{1}{3}\left(a+b+c\right)\le1\).
Đẳng thức xảy ra khi a = b = c = 1.
Theo đề ra, ta có:
\(a^2+b^2+c^2\)
\(=\left(a+b+c\right)\left(a^2+b^2+c^2\right)\)
\(=a^3+b^3+c^3+a^2b+b^2c+c^2a+ab^2+bc^2+ca^2\)
Theo BĐT Cô-si:
\(\left\{{}\begin{matrix}a^3+ab^2\ge2a^2b\\b^3+bc^2\ge2b^2c\\c^3+ca^2\ge2c^2a\end{matrix}\right.\Rightarrow a^2+b^2+c^2\ge3\left(a^2b+b^2c+c^2a\right)\)
Do vậy \(M\ge14\left(a^2+b^2+c^2\right)+\dfrac{3\left(ab+bc+ac\right)}{a^2+b^2+c^2}\)
Ta đặt \(a^2+b^2+c^2=k\)
Luôn có \(3\left(a^2+b^2+c^2\right)\ge\left(a+b+c\right)^2=1\)
Vì thế nên \(k\ge\dfrac{1}{3}\)
Khi đấy:
\(M\ge14k+\dfrac{3\left(1-k\right)}{2k}=\dfrac{k}{2}+\dfrac{27k}{2}+\dfrac{3}{2k}-\dfrac{3}{2}\ge\dfrac{1}{3}.\dfrac{1}{2}+2\sqrt{\dfrac{27k}{2}.\dfrac{3}{2k}}-\dfrac{3}{2}=\dfrac{23}{3}\)
\(\Rightarrow Min_M=\dfrac{23}{3}\Leftrightarrow a=b=c=\dfrac{1}{3}\).
Giải
a, 2A+3B=0 <=> \(\dfrac{10}{2m+1}+\dfrac{12}{2m-1}=0\)
<=>10(2m-1)+ 12(2m+1) =0
<=> 44m +2 =0
<=> m=-1/22
b, AB= A+B <=> \(\dfrac{20}{\left(2m-1\right)\left(2m+1\right)}=\dfrac{5}{2m+1}+\dfrac{4}{2m-1}\)
<=> 20 = 5(2m -1) + 4(2m+1)
<=> 20 = 18m - 1
<=> m=7/6
\(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=\dfrac{1}{2016}\)
\(\Rightarrow\dfrac{bc+ac+bc}{abc}=\dfrac{1}{2016}\)
\(\Rightarrow\dfrac{bc+ac+ab}{abc}=\dfrac{1}{a+b+c}\)
\(\Rightarrow\left(ab+bc+ca\right)\left(a+b+c\right)=abc\)
\(\Rightarrow ab\left(a+b\right)+bc\left(b+c\right)+ca\left(c+a\right)+3abc=abc\)
\(\Rightarrow ab\left(a+b\right)+bc\left(b+c\right)+ca\left(c+a\right)+2abc=0\)
\(\Rightarrow\left(a+b\right)\left(b+c\right)\left(c+a\right)=0\)
\(\Rightarrow a=-b\) hay \(b=-c\) hay \(c=-a\)
-Vậy trong ba số a,b,c tồn tại 2 số đối nhau.
\(a\ge2b\Rightarrow\dfrac{a}{b}\ge2\)
\(A=\dfrac{a}{b}+\dfrac{b}{a}=\dfrac{a}{4b}+\dfrac{b}{a}+\dfrac{3}{4}.\dfrac{a}{b}\ge2\sqrt{\dfrac{ab}{4ab}}+\dfrac{3}{4}.2=\dfrac{5}{2}\)
\(A_{min}=\dfrac{5}{2}\) khi \(a=2b\)
-Đề thiếu?
e sửa r ạ