Question

Tìm GTNN của \(A=\dfrac{5}{\left(x+y\right)^2+\left(y+z\right)^4+\left(z+x\right)^6}\)

Answer 1

p/s :viết đề cho cẩn thận nha bạn,câu nào cx sai tek =="

\(A=\dfrac{5}{\left(x+y\right)^2+\left(y+z\right)^4+\left(z+x\right)^6+1}\)

Ta có:

\(\left\{{}\begin{matrix}\left(x+y\right)^2\ge0\\\left(y+z\right)^4\ge0\\\left(z+x\right)^6\ge0\end{matrix}\right.\) \(\Leftrightarrow\left(x+y\right)^2+\left(y+z\right)^4+\left(z+x\right)^6\ge0\)

\(\Leftrightarrow\left(x+y\right)^2+\left(y+z\right)^4+\left(z+x\right)^6+1\ge1\)

\(A=\dfrac{5}{\left(x+y\right)^2+\left(y+z\right)^4+\left(z+x\right)^6}\le5\)
Dấu "=" xảy ra khi:

\(\left\{{}\begin{matrix}\left(x+y\right)^2=0\\\left(y+z\right)^4=0\\\left(z+x\right)^6=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=-y\\-z=y\\x=-z\end{matrix}\right.\)

\(\Leftrightarrow x=-z=-y\)