\(x^2-2\left(m+1\right)x+m^2+3=0\)
\(\Delta'=\left[-\left(m+1\right)\right]^2-\left(m^2+3\right)\)
\(=m^2+2m+1-m^2-3\)
\(=2m-2\)
Để phương trình có nghiệm thì \(\Delta'\ge0\)
\(\Leftrightarrow2m-2\ge0\)
\(\Leftrightarrow2m\ge2\)
\(\Leftrightarrow m\ge1\)
Theo hệ thức Vi-ét, ta có:
\(\left\{{}\begin{matrix}x_1+x_2=-\dfrac{b}{a}=2\left(m+1\right)\\x_1x_2=\dfrac{c}{a}=m^2+3\end{matrix}\right.\)
\(A=x_1+x_1x_2+x_2=x_1+x_2+x_1x_2\)
\(=2\left(m+1\right)+m^2+3\)
\(=m^2+2m+2+3\)
\(=\left(m+1\right)^2+4\ge4\)
Lại có: \(m\ge1\)
\(\Rightarrow min_A=\left(1+1\right)^2+4=8\)
\(x^2-2x+m-3=0\)
\(\Delta'=\left(-1\right)^2-\left(m-3\right)\)
\(=1-m+3\)
\(=4-m\)
Phương trình có nghiệm khi \(\Delta'\ge0\)
\(\Leftrightarrow4-m\ge0\)
\(\Leftrightarrow m\le4\)
Theo hệ thức Vi-ét, ta có:
\(\left\{{}\begin{matrix}x_1+x_2=2\\x_1x_2=m-3\end{matrix}\right.\)
\(B=x_1^2x_2+x_1x_2^2\)
\(=x_1x_2\left(x_1+x_2\right)\)
\(=\left(m-3\right).2\)
\(=2m-6\)
Lại có \(m\le4\)
\(\Rightarrow max_B=2.4-6=2\)
\(C=x_1^3x_2+x_1x_2^3\)
\(=x_1x_2\left(x_1^2+x_2^2\right)\)
\(=x_1x_2\left[\left(x_1+x_2\right)^2-2x_1x_2\right]\)
\(=\left(m-3\right)\left[2^2-2\left(m-3\right)\right]\)
\(=\left(m-3\right)\left(4-2m+6\right)\)
\(=\left(m-3\right)\left(10-2m\right)\)
\(=10m-2m^2-30+6m\)
\(=-2m^2+16m-30\)
\(=-2\left(m^2-8m+15\right)\)
\(=-2\left(m^2-8m+16-1\right)\)
\(=-2\left[\left(m-4\right)^2-1\right]\)
\(=-2\left(m-4\right)^2+2\le2\)
\(\Rightarrow max_C=2\Leftrightarrow m=4\)