Question 3: Find all real values of parameter m so that the function \(y=x^{3}-3 mx^{2}+(m-1) x+2\) has a maximum and a minimum and the extreme points of the graph of the function have positive coordinates
We have \(y^{\prime}=3 x^{2}-6 m x+m-1\).
The function has a maximum and a minimum if and only if PT y’ = 0 has two distinct solutions
This is equivalent \(\Delta^{\prime}=9 m^{2}-3(m-1)>0 \Leftrightarrow 3 m^{2}-m+1>0 \text { (true for all } \left. m\right)\)
Two extreme points have positive horizontal \(\Leftrightarrow\left\{\begin{array}{l} S>0 \\ P>0 \end{array} \Leftrightarrow\left\{\begin{array}{l} 2 m>0 \\ \ frac{m-1}{3}>0 \end{array} \Leftrightarrow m>1\right.\right.\)So the required values of m are m > 1.
===============