Question 36: Given the function (y=x^3-(m+1)x^2-(m^2-2m)x+2020). Find m so that the function is inverse on the interval (0, 1)

Question 36: Given the function \(y=x^3-(m+1)x^2-(m^2-2m)x+2020\). Find m so that the function is inverse on the interval (0, 1)

A. \(1 < m < \frac{3}{2}\)

B. \(1 \leq m \leq \frac{3}{2}\)

C. \(-\frac{3}{2} \leq m \leq -1\)

D. \(-1 \leq m \leq \frac{3}{2}\)

We have:

\(\begin{array}{l} y^{\prime}=3 x^{2}-2(m+1) x-\left(m^{2}-2 m\right) \\ y^ {\prime}=0 \Leftrightarrow x=m ; x=\frac{m-2}{3} \end{array}\)

The given function is inverse (0;1) if and only if:

\(\left[\begin{array}{l}m\leq0<1\leq\frac{m-2}{3}\\\frac{m-2}{3}\leq0<1\leqm\end{array}\Leftrightarrow1\leqm\leq\frac{3}{2}\right\)

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