Đáp án:
\[m < \dfrac{4}{5}\]
Giải thích các bước giải:
Hàm số \(y = a\,{x^2} + bx + c < 0,\,\,\,\forall x \in R \Leftrightarrow \left\{ \begin{array}{l}
a < 0\\
\Delta < 0
\end{array} \right.\)
Ta có:
\(\begin{array}{l}
f\left( x \right) < 0,\,\,\,\forall x \in R\\
\Leftrightarrow \left( {m - 1} \right){x^2} - 2mx + m - 4 < 0,\,\,\,\,\forall x \in R\\
\Leftrightarrow \left\{ \begin{array}{l}
m - 1 < 0\\
\Delta ' < 0
\end{array} \right.\\
\Leftrightarrow \left\{ \begin{array}{l}
m < 1\\
{m^2} - \left( {m - 1} \right)\left( {m - 4} \right) < 0
\end{array} \right.\\
\Leftrightarrow \left\{ \begin{array}{l}
m < 1\\
{m^2} - \left( {{m^2} - 5m + 4} \right) < 0
\end{array} \right.\\
\Leftrightarrow \left\{ \begin{array}{l}
m < 1\\
5m - 4 < 0
\end{array} \right.\\
\Leftrightarrow \left\{ \begin{array}{l}
m < 1\\
m < \dfrac{4}{5}
\end{array} \right.\\
\Leftrightarrow m < \dfrac{4}{5}
\end{array}\)
Vậy \(m < \dfrac{4}{5}\)
