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Câu hỏi: Có tất cả bao nhiêu giá trị nguyên của tham số $m$ để hàm số $y=\left| 3{{x}^{4}}-m{{x}^{3}}+6{{x}^{2}}+m-3 \right|$ đồng biến trên khoảng $\left( 0;+\infty \right)$ ?
A. 5.
B. 6.
C. 4.
D. 7.
A. 5.
B. 6.
C. 4.
D. 7.
$f\left( x \right)=3{{x}^{4}}-m{{x}^{3}}+6{{x}^{2}}+m-3$
Do $\underset{x\to +\infty }{\mathop{\lim }} f\left( x \right)=+\infty >0$ nên $\left| f\left( x \right) \right|$ đồng biến trên $\left( 0;+\infty \right)$
$\left\{\begin{array}{l}f(x)>0 \\ f^{\prime}(x)>0\end{array}, \forall x \in(0 ;+\infty) \Leftrightarrow\left\{\begin{array}{l}f(x) \geq 0 \\ f^{\prime}(x) \geq 0\end{array}, \forall x \in(0 ;+\infty)\right.\right.$
$\Leftrightarrow\left\{\begin{array}{l}m-3 \geq 0 \\ 12 x^3-3 m x^2+12 x \geq 0\end{array}, \forall x \in(0 ;+\infty) \Leftrightarrow\left\{\begin{array}{l}m \geq 3 \\ m \leq 4 x+\dfrac{4}{x}\end{array}, \forall x \in(0 ;+\infty)\right.\right.$
$\Leftrightarrow\left\{\begin{array}{l}m \geq 3 \\ m \leq \min _{(0 ;+\infty)}\left(4 x+\dfrac{4}{x}\right)\end{array}\right.$ $\Leftrightarrow\left\{\begin{array}{l}m \geq 3 \\ m \leq 8\end{array} \Leftrightarrow 3 \leq m \leq 8\right.$
Do $m\in Z\Rightarrow m\in \left\{ 3,4,5,6,7,8 \right\}.$
Do $\underset{x\to +\infty }{\mathop{\lim }} f\left( x \right)=+\infty >0$ nên $\left| f\left( x \right) \right|$ đồng biến trên $\left( 0;+\infty \right)$
$\left\{\begin{array}{l}f(x)>0 \\ f^{\prime}(x)>0\end{array}, \forall x \in(0 ;+\infty) \Leftrightarrow\left\{\begin{array}{l}f(x) \geq 0 \\ f^{\prime}(x) \geq 0\end{array}, \forall x \in(0 ;+\infty)\right.\right.$
$\Leftrightarrow\left\{\begin{array}{l}m-3 \geq 0 \\ 12 x^3-3 m x^2+12 x \geq 0\end{array}, \forall x \in(0 ;+\infty) \Leftrightarrow\left\{\begin{array}{l}m \geq 3 \\ m \leq 4 x+\dfrac{4}{x}\end{array}, \forall x \in(0 ;+\infty)\right.\right.$
$\Leftrightarrow\left\{\begin{array}{l}m \geq 3 \\ m \leq \min _{(0 ;+\infty)}\left(4 x+\dfrac{4}{x}\right)\end{array}\right.$ $\Leftrightarrow\left\{\begin{array}{l}m \geq 3 \\ m \leq 8\end{array} \Leftrightarrow 3 \leq m \leq 8\right.$
Do $m\in Z\Rightarrow m\in \left\{ 3,4,5,6,7,8 \right\}.$
Đáp án B.