Question 50: Let \(x_1;x_2\) be the two extreme points of the function \(y=x^{3}-3 mx^{2}+3\left(m^{2}-1\right) xm^{3}+m\) . Find all values of the real parameter m so that \(x_{1}^{2}+x_{2}^{2}-x_{1} x_{2}=7\)
\(y^{\prime}=3 x^{2}-6 m x+3\left(m^{2}-1\right)\)
The function always has a maximum for every m
According to Viet’s theorem \(\left\{\begin{array}{l} x_{1}+x_{2}=2 m \\ x_{1} \cdot x_{2}=m^{2}-1 \end{array }\right.\)
\(x_{1}^{2}+x_{2}^{2}-x_{1} x_{2}=7 \Leftrightarrow(2 m)^{2}-3\left(m^{2} -1\right)=7 \Leftrightarrow m=\pm 2\)
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