Từ \(xyzt=1\) ta có: \(\dfrac{1}{x^3\left(yz+zt+ty\right)}=\dfrac{xyzt}{x^3\left(yz+zt+ty\right)}=\dfrac{yzt}{x^2\left(yz+zt+ty\right)}\)
Đánh giá tương tự ta có:
\(pt\Leftrightarrow\dfrac{yzt}{x^2\left(yz+zt+ty\right)}+\dfrac{xzt}{y^2\left(xz+zt+tx\right)}+\dfrac{xyt}{z^2\left(xy+yt+tx\right)}+\dfrac{xyz}{t^2\left(xy+yz+zx\right)}\ge3\left(yzt+xzt+xyt+xyz\right)=3yzt+3xzt+3xyt+3xyz\)
Ta sẽ chứng minh:
\(\dfrac{yzt}{x^2\left(yz+zt+ty\right)}\ge3yzt\). Cộng theo vế rồi suy ra đpcm
T gần đi học r,có gì tối về giải full cho
Áp dụng cauchy-schwarz:
\(VT=\sum\dfrac{\dfrac{1}{x^2}}{\dfrac{1}{y}+\dfrac{1}{z}+\dfrac{1}{t}}\ge\dfrac{\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}+\dfrac{1}{t}\right)^2}{3\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}+\dfrac{1}{t}\right)}=VF\)
\(VT=\dfrac{1}{x^3\left(yz+zt+ty\right)}+\dfrac{1}{y^3\left(xz+zt+tx\right)}+\dfrac{1}{z^3\left(xy+yt+tx\right)}+\dfrac{1}{t^3\left(xy+yz+xz\right)}\)
\(=\dfrac{\dfrac{1}{x^2}}{xyz+xzt+xyt}+\dfrac{\dfrac{1}{y^2}}{xyz+yzt+txy}+\dfrac{\dfrac{1}{z^2}}{xyz+yzt+ztx}+\dfrac{\dfrac{1}{t^2}}{xyt+yzt+txz}\)
\(=\dfrac{\dfrac{1}{x^2}}{\dfrac{xyz}{xyzt}+\dfrac{xzt}{xyzt}+\dfrac{xyt}{xyzt}}+\dfrac{\dfrac{1}{y^2}}{\dfrac{xyz}{xyzt}+\dfrac{yzt}{xyzt}+\dfrac{txy}{xyzt}}+\dfrac{\dfrac{1}{z^2}}{\dfrac{xyz}{xyzt}+\dfrac{yzt}{xyzt}+\dfrac{ztx}{xyzt}}+\dfrac{\dfrac{1}{t^2}}{\dfrac{xyt}{xyzt}+\dfrac{yzt}{xyzt}+\dfrac{txz}{xyzt}}\)
\(=\dfrac{\dfrac{1}{x^2}}{\dfrac{1}{t}+\dfrac{1}{y}+\dfrac{1}{z}}+\dfrac{\dfrac{1}{y^2}}{\dfrac{1}{t}+\dfrac{1}{x}+\dfrac{1}{z}}+\dfrac{\dfrac{1}{z^2}}{\dfrac{1}{t}+\dfrac{1}{x}+\dfrac{1}{y}}+\dfrac{\dfrac{1}{t^2}}{\dfrac{1}{z}+\dfrac{1}{x}+\dfrac{1}{y}}\)
\(\ge\dfrac{\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}+\dfrac{1}{t}\right)^2}{3\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}+\dfrac{1}{t}\right)}=\dfrac{1}{3}\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}+\dfrac{1}{t}\right)=VP\)
