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Giải bài 24 trang 52 SBT toán 10 - Cánh diều

Tìm \(m\) để tam thức \(f\left( x \right) = - {x^2} - 2x + m - 12\) không dương với mọi \(x \in \mathbb{R}\)

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Đề bài

Tìm \(m\) để tam thức \(f\left( x \right) =  - {x^2} - 2x + m - 12\) không dương với mọi \(x \in \mathbb{R}\)

Phương pháp giải - Xem chi tiết

Tam thức \(f\left( x \right) = a{x^2} + bx + c \le 0\;\forall x \in \mathbb{R} \Leftrightarrow \left\{ \begin{array}{l}a < 0\\\Delta  \le 0\end{array} \right.\)

Lời giải chi tiết

Hàm số \(f\left( x \right) =  - {x^2} - 2x + m - 12 \le 0\;\forall x \in \mathbb{R} \Leftrightarrow \left\{ \begin{array}{l}a < 0\\\Delta  \le 0\end{array} \right.\)  (*)Mà \(a =  - 1 < 0\) nên\(\left( * \right) \Leftrightarrow \Delta  = {\left( { - 2} \right)^2} - 4.\left( { - 1} \right).\left( {m - 12} \right) \le 0 \Leftrightarrow 4m - 44 \le 0 \Leftrightarrow m \le 11\)Vậy \(m \le 11\) thì tam thức đó không dương với mọi \(x \in \mathbb{R}\).

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