cho x,y,z>0 và \(x^2+y^2+z^2=3\)
tìm Min \(A=\frac{y^2z^2}{x\left(y^2+z^2\right)}+\frac{x^2y^2}{z\left(x^2+y^2\right)}+\frac{z^2x^2}{y\left(z^2+x^2\right)}\)
giải hộ mk nhé mk đang cần gấp
Cho x,y,z>0 thỏa mãn xyz=1. Tìm min \(P=\frac{x^2\left(y+z\right)}{y\sqrt{y}+2z\sqrt{z}}+\frac{y^2\left(z+x\right)}{z\sqrt{z}+2x\sqrt{x}}+\frac{z^2\left(x+y\right)}{x\sqrt{x}+2y\sqrt{y}}\)
bạn vào trang này nhé có bài như thến này đấy
//123doc.org//document/3173507-ren-luyen-chuyen-de-tim-maxmin-on-thi-thpt-quoc-gia.htm
tính diện tích hình vẽ dưới đây
đặt \(A=\frac{\sqrt{yz}}{x+3\sqrt{yz}}+\frac{\sqrt{zx}}{y+3\sqrt{zx}}+\frac{\sqrt{xy}}{z+3\sqrt{xy}}\)
\(\Rightarrow1-3A=\frac{x}{x+3\sqrt{yz}}+\frac{y}{y+3\sqrt{zx}}+\frac{z}{z+3\sqrt{xy}}\)
\(\ge\frac{x}{x+\frac{3}{2}\left(y+z\right)}+\frac{y}{y+\frac{3}{2}\left(z+x\right)}+\frac{z}{z+\frac{3}{2}\left(x+y\right)}\)
\(=\frac{2x}{2x+3\left(y+z\right)}+\frac{2y}{2y+3\left(z+x\right)}+\frac{2z}{2z+3\left(x+y\right)}\)
\(=\frac{2x^2}{2x^2+3xy+3xz}+\frac{2y^2}{2y^2+3yz+3xy}+\frac{2z^2}{2z^2+3zx+3yz}\)
\(\ge\frac{2\left(x+y+z\right)^2}{2\left(x^2+y^2+z^2\right)+6\left(xy+yz+zx\right)}=\frac{2\left(x+y+z\right)^2}{2\left(x+y+z\right)^2+2\left(xy+yz+zx\right)}\)
\(\ge\frac{2\left(x+y+z\right)^2}{2\left(x+y+z\right)^2+\frac{2}{3}\left(x+y+z\right)^2}=\frac{2\left(x+y+z\right)^2}{\frac{8}{3}\left(x+y+z\right)^2}=\frac{3}{4}\)
\(\Rightarrow1-3A\ge\frac{3}{4}\Rightarrow A\le\frac{3}{4}\left(Q.E.D\right)\)
cho x,y,z dương;
\(3\left(x^4+y^4+z^4\right)-7\left(x^2+y^2+z^2\right)+12=0\)
tìm min
\(\frac{x^2}{y+2z}+\frac{y^2}{z+2x}+\frac{z^2}{x+2y}\)
Cần mọi người giúp bài Bất đẳng thức - Diễn Đàn MathScope
thực hiện phép tính
a,\(x^3+\left[\frac{x\left(2y^3-x^3\right)}{x^3+y^3}\right]^3-\left[\frac{y\left(2x^3-y^3\right)}{x^3+y^3}\right]^3\)
b,\(\frac{\frac{x\left(x+y\right)}{x-y}+\frac{x\left(x+z\right)}{x-z}}{1+\frac{\left(y-z\right)^2}{\left(x-y\right)\left(x-z\right)}}+\frac{\frac{y\left(y+z\right)}{y-z}+\frac{y\left(y+x\right)}{y-x}}{1+\frac{\left(z-x\right)^2}{\left(y-z\right)\left(y-x\right)}}+\frac{\frac{z\left(z+x\right)}{z-x}+\frac{z\left(z+y\right)}{z-y}}{1+\frac{\left(x-y\right)^2}{\left(z-x\right)\left(z-y\right)}}\)
c,\(\left[\frac{y+z-2x}{\frac{\left(y-z\right)^3}{y^3-z^3}+\frac{\left(x-y\right)\left(x-z\right)}{y^2+yz+z^2}}+\frac{z+x-2y}{\frac{\left(z-x\right)^3}{z^3-x^3}+\frac{\left(y-z\right)\left(y-x\right)}{z^2+xz+x^2}}+\frac{x+y-2z}{\frac{\left(x-y\right)^3}{x^3-y^3}+\frac{\left(z-x\right)\left(z-y\right)}{x^2+xy+y^2}}\right]:\frac{1}{x+y+z}\)
Cho x,y,z>0 va xyz=1. Tim Min cua \(P=\frac{x^2\left(y+z\right)}{y\sqrt{y}+2z\sqrt{z}}+\frac{y^2\left(z+x\right)}{z\sqrt{z}+2x\sqrt{x}}+\frac{z^2\left(x+y\right)}{x\sqrt{x}+2y\sqrt{y}}\)
Cho a,b,c>0; a+b+c=3/4. Tìm min
\(M=6\left(x^2+y^2+z^2\right)+10\left(xy+yz+zx\right)+2.\left(\frac{1}{2x+y+z}+\frac{1}{x+2y+z}+\frac{1}{x+y+2z}\right)\)
\(M=5\left(x+y+z\right)^2+\left(x^2+y^2+z^2\right)+2.\left(\frac{1}{2x+y+z}+\frac{1}{x+2y+z}+\frac{1}{x+y+2z}\right)\)
Áp dụng BĐT Cauchy-schwarz ta có:
\(M\ge5.\left(\frac{3}{4}\right)^2+\frac{\left(x+y+z\right)^2}{3}+2.\frac{\left(1+1+1\right)^2}{4\left(x+y+z\right)}=5.\frac{9}{16}+\frac{\frac{9}{16}}{3}+2.\frac{9}{\frac{4.3}{4}}=9\)
Dấu " = " xảy ra <=> a=b=c=1/4 ( cái này bạn tự giải rõ nhé)
tự hỏi tự trả lời ?
Lại còn " cái này bn tự giải rõ nhé " :v
Cho x,y,z>0. Cmr \(\frac{x^3}{\left(y+2z\right)^2}+\frac{y^3}{\left(z+2x\right)^2}+\frac{z^3}{\left(x+2y\right)^2}\ge\frac{2\left(x+y+z\right)}{9}\)
Cho \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=0\). Tính \(C=\left(\frac{x^2+y^2}{x^2y^2}-z^2\right)\left(\frac{y^2+z^2}{y^2z^2}-x^2\right)\left(\frac{z^2+x^2}{z^2x^2}-y^2\right)\)
Cho x + y + z khác 0 ; x = y + z . Chứng minh rằng :
\(\frac{\left(xy+yz+zx\right)^2-\left(x^2y^2+y^2z^2+z^2x^2\right)}{x^2+y^2+z^2}:\frac{\left(x+y+z\right)^2}{x^2+y^2+z^2}=yz\)