Question 48: Find all real values of parameter m so that the function y = x3-3×2+3mx+1 has the extreme points less than 2
We have y’ = 3x2-6x+3m
Ask the problem when y’ = 0 has two distinct solutions xfirst < x2 < 2
\(\begin{array}{l}
\Leftrightarrow \left\{ \begin{array}{l}
\Delta ‘ = 9 – 9m > 0\\
\left( {{x_1} – 2} \right) + \left( {{x_2} – 2} \right) < 0\\
\left( {{x_1} – 2} \right)\left( {{x_2} – 2} \right) > 0
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
m < 1\\
{x_1} + {x_2} < 4\\
{x_1}{x_2} – 2\left( {{x_1} + {x_2}} \right) + 4 > 0
\end{array} \right.\\
\Leftrightarrow \left\{ \begin{array}{l}
m < 1\\
2 < 4\\
m – 2.2 + 4 > 0
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
m < 1\\
m > 0
\end{array} \right. \Leftrightarrow 0 < m < 1
\end{array}\)
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