Question

[N.T.M.D]

Cho hệ ptrình tham số m

\(\hept{\begin{cases}x+my=3\\mx+y=2m+1\end{cases}}\)

Biết hệ có nghiệm duy nhất (x,y).Tìm giá trị nhỏ nhất P=\(x^2\)+\(3y^2\)

Answer 1

[Nguyễn Việt Lâm]

\(\left\{{}\begin{matrix}x+my=3\\m^2x+my=2m^2+m\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}x+my=3\\\left(m^2-1\right)x=2m^2+m-3\end{matrix}\right.\)

\(\Rightarrow\left\{{}\begin{matrix}x+my=3\\x=\dfrac{2m+3}{m+1}\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=\dfrac{2m+3}{m+1}\\y=\dfrac{1}{m+1}\end{matrix}\right.\)

\(P=\left(\dfrac{2m+3}{m+1}\right)^2+\dfrac{3}{\left(m+1\right)^2}=\left(2+\dfrac{1}{m+1}\right)^2+\dfrac{3}{\left(m+1\right)^2}\)

\(=4+\dfrac{4}{m+1}+\dfrac{4}{\left(m+1\right)^2}=\left(\dfrac{2}{m+1}+1\right)^2+3\ge3\)

\(P_{min}=3\) khi \(m=-3\)