Toán Lớp 11: tìm m để phường trình 5cos(5x+1) =2m²-m-5
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Toán Lớp 11: tìm m để phường trình 5cos(5x+1) =2m²-m-5
Comments ( 1 )
5.\cos \left( {5x + 1} \right) = 2{m^2} – m – 5\\
\Leftrightarrow \cos \left( {5x + 1} \right) = \dfrac{{2{m^2} – m – 5}}{5}\\
Do: – 1 \le \cos \left( {5x + 1} \right) \le 1\\
\Leftrightarrow – 1 \le \dfrac{{2{m^2} – m – 5}}{5} \le 1\\
\Leftrightarrow \left\{ \begin{array}{l}
\dfrac{{2{m^2} – m – 5}}{5} + 1 \ge 0\\
\dfrac{{2{m^2} – m – 5}}{5} – 1 \le 0
\end{array} \right.\\
\Leftrightarrow \left\{ \begin{array}{l}
2{m^2} – m – 5 + 5 \ge 0\\
2{m^2} – m – 10 \le 0
\end{array} \right.\\
\Leftrightarrow \left\{ \begin{array}{l}
m\left( {2m – 1} \right) \ge 0\\
\left( {2m – 5} \right)\left( {m + 2} \right) \le 0
\end{array} \right.\\
\Leftrightarrow \left\{ \begin{array}{l}
\left[ \begin{array}{l}
m \ge \dfrac{1}{2}\\
m \le 0
\end{array} \right.\\
– 2 \le m \le \dfrac{5}{2}
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
\dfrac{1}{2} \le m \le \dfrac{5}{2}\\
– 2 \le m \le 0
\end{array} \right.\\
Vậy\,m \in \left[ { – 2;0} \right] \cup \left[ {\dfrac{1}{2};\dfrac{5}{2}} \right]
\end{array}$