cho a,b,c>0 thoa man a+b+c=3. Tim GTLN cua P=\(a\sqrt{b}+b\sqrt{c}+c\sqrt{a}-\sqrt{abc}\)
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Cho a,b,c>0 thoa man a+b+c=3. Tìm GTLN cua \(a\sqrt{b}+b\sqrt{c}+c\sqrt{a}-\sqrt{abc}\)
cho a b c la cac so thuc ko am thoa man a+b+c=3. tim GTLN cua K=\(\sqrt{12a+\left(b-c\right)^2}+\sqrt{12b+\left(a-c\right)^2}+\sqrt{12c+\left(a-b\right)^2}\)
Bài này hay:)
c = min {a,b,c}. Đặt
\(a-c=x;b-c=y\Rightarrow x,y\ge0\) và x + y = a + b - 2c \(=3-3c\le3\)
\(\Rightarrow a-b=x-y;c=\frac{3-x-y}{3}\)
\(a=x+c=x+\frac{3-x-y}{3}=\frac{2x-y+3}{3}\)
\(b=y+c=\frac{2y-x+3}{3}\)
Như vậy: \(K=\sqrt{4\left(2x-y+3\right)+y^2}+\sqrt{4\left(2y-x+3\right)+x^2}+\sqrt{4\left(3-x-y\right)+\left(x-y\right)^2}\)
\(=\sqrt{y^2-4y+8x+12}+\sqrt{x^2-4x+8y+12}+\sqrt{4\left(3-x-y\right)+\left(x-y\right)^2}\)
Giờ em đang bận, tối em làm tiếp!
\(12a+\left(b-c\right)^2=4a\left(a+b+c\right)+b^2-2bc+c^2\)
\(=4a^2+b^2+c^2+4ab+4ac+2bc-4bc\)
\(=\left(2a+b+c\right)^2-4bc\le\left(2a+b+c\right)^2\)
\(\Rightarrow\sqrt{12a+\left(b-c\right)^2}\le2a+b+c\)
Tương tự: \(\sqrt{12b+\left(a-c\right)^2}\le a+2b+c\); \(\sqrt{12c+\left(a-b\right)^2}\le a+b+2c\)
Cộng vế với vế:
\(K\le4\left(a+b+c\right)=12\)
Dấu "=" xảy ra khi \(\left(a;b;c\right)=\left(0;0;3\right)\) và các hoán vị
Cho a,b,c duong thoa man a +b+c=1
Tim GTLN P=\(\sqrt{\dfrac{ab}{ab+c}}+\sqrt{\dfrac{bc}{bc+a}}+\sqrt{\dfrac{ac}{ac+b}}\)
Sử dụng AM-GM, ta có
\(P=\sum\sqrt{\dfrac{ab}{ab+c}}=\sum\sqrt{\dfrac{ab}{ab+c\left(a+b+c\right)}}=\sum\sqrt{\dfrac{ab}{\left(c+b\right)\left(c+a\right)}}\le\dfrac{1}{2}\sum\dfrac{a}{c+b}+\dfrac{b}{c+a}=\dfrac{3}{2}\)
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cho a,b,c la do dai 3 canh cua mot tam giac thoa man dieu kien \(\sqrt{a+b-c}+\sqrt{b+c-a}+\sqrt{c+a-b}=\sqrt{a}+\sqrt{b}+\sqrt{c}\)
chung minh a,b,c la 3 canh cua mot tam giac deu
Tim cac so nguyen duong a, b, c thoa man: \(\left\{{}\begin{matrix}\sqrt{a-b+c}=\sqrt{a}-\sqrt{b}+\sqrt{c}\\\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=1\end{matrix}\right.\)
cho a;b;c thoa man a+b+c=4(a;b;c>0)
c/m \(\sqrt{a+b}+\sqrt{b+c}+\sqrt{a+c}>4\)
tim tat ca cac so duong a,b,c thoa man dieu kien \(\left\{{}\begin{matrix}\sqrt{a}+\sqrt{b}+\sqrt{c}=6\\\frac{1}{\sqrt{a}}+\frac{4}{\sqrt{b}}+\frac{9}{\sqrt{c}}=6\end{matrix}\right.\)
Áp dụng BĐT Cauchy-Schwarz:
\(\frac{1^2}{\sqrt{a}}+\frac{2^2}{\sqrt{b}}+\frac{3^2}{\sqrt{c}}\ge\frac{\left(1+2+3\right)^2}{\sqrt{a}+\sqrt{b}+\sqrt{c}}=\frac{36}{6}=6\)
Dấu "=" xảy ra khi và chỉ khi \(\left\{{}\begin{matrix}\frac{1}{\sqrt{a}}=\frac{2}{\sqrt{b}}=\frac{3}{\sqrt{c}}\\\sqrt{a}+\sqrt{b}+\sqrt{c}=6\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}\sqrt{a}=1\\\sqrt{b}=2\\\sqrt{c}=3\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a=1\\b=4\\c=9\end{matrix}\right.\)
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1 cho 3 so thuc duong thoa man x^2010+y^2010+z^20103 tim gia tri lon nhat cua x^2+y^2+z^22 cho a;b;c duong c/m frac{1}{a}+frac{1}{b}+frac{1}{c}hoac3left(frac{1}{a+2b}+frac{1}{b+2c}+frac{1}{c+2a}right)3 tim gia tri nho nhat cua sqrt{a^2+ab+b^2}+sqrt{b^2+bc+c^2}+sqrt{c^2+ac+a^2} voi a+b+c14 cho a;b;c;d va A;B;C;D la cac so duong thoa man frac{a}{A}frac{b}{B}frac{c}{C}frac{d}{D}C/ M sqrt{aA}+sqrt{bB}+sqrt{cC}+sqrt{dD}sqrt{left(a+b+c+dright)left(A+B+C+Dright)}5 tim gia tr...
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1 cho 3 so thuc duong thoa man x^2010+y^2010+z^2010=3 tim gia tri lon nhat cua x^2+y^2+z^2
2 cho a;b;c duong c/m \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}>hoac=3\left(\frac{1}{a+2b}+\frac{1}{b+2c}+\frac{1}{c+2a}\right)\)
3 tim gia tri nho nhat cua \(\sqrt{a^2+ab+b^2}+\sqrt{b^2+bc+c^2}+\sqrt{c^2+ac+a^2}\) voi a+b+c=1
4 cho a;b;c;d va A;B;C;D la cac so duong thoa man \(\frac{a}{A}=\frac{b}{B}=\frac{c}{C}=\frac{d}{D}\)C/ M \(\sqrt{aA}+\sqrt{bB}+\sqrt{cC}+\sqrt{dD}=\sqrt{\left(a+b+c+d\right)\left(A+B+C+D\right)}\)
5 tim gia tri lon nhat cua \(\frac{yz\sqrt{x-1}+xz\sqrt{y-2}+xy\sqrt{z-3}}{xyz}\)
6 phan tich da thuc thanh nhan tu \(y-5x\sqrt{y}+6x^2\)
7 cho x;y;z>0 xy+yz+xz=1 tinh \(x\sqrt{\frac{\left(1+y^2\right)\left(1+z^2\right)}{1+x^2}}+y\sqrt{\frac{\left(1+x^2\right)\left(1+z^2\right)}{1+y^2}}+z\sqrt{\frac{\left(1+x^2\right)\left(1+y^2\right)}{1+z^2}}\)
8 cho a;b;c >0 c/m \(\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}
pn oi nhieu the nay ai ma giai cho het dc
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bài lớp mấy mà nhìn ghê quá zật bạn..................Nhìu quá
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Cho a,b,c la 3 so thuc thoa man :a+b+c=\(\sqrt{a}+\sqrt{b}+\sqrt{c}=2\)
C/m \(\dfrac{\sqrt{a}}{1+a}+\dfrac{\sqrt{b}}{1+b}+\dfrac{\sqrt{c}}{1+c}=\dfrac{2}{\sqrt{\left(1+a\right)\left(1+b\right)\left(1+b\right)}}\)
từ giả thiết ,ta có:\(\left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)^2=4\)\(\Leftrightarrow a+b+c+2\left(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\right)=4\)
\(\Leftrightarrow\sqrt{ab}+\sqrt{bc}+\sqrt{ac}=1\)---> thay 1= vào ...
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