Có bao nhiêu số nguyên $\Large y\in (-20; 20)$ thỏa mãn $\Large 2+\log

Có bao nhiêu số nguyên $\Large y\in (-20; 20)$ thỏa mãn $\Large 2+\log_{\sqrt{3}}\left(3x^2+1\right)\leq \log_{\sqrt{3}}\left(yx^2-6x+2y\right)$ với mọi $\Large x\in \mathbb{R}$?

• A.

  9.

• B.

  11.

• C.

  10.

• D.

  8.

Chọn C

Ta có: $\Large 2+\log_{\sqrt{3}}\left(3x^2+1\right)\leq \log_{\sqrt{3}}\left(yx^2-6x+2y\right)$ (1) với mọi $\Large x\in \mathbb{R}$.

ĐKXĐ: $\Large yx^2-6x+2y > 0$, $\Large \forall x\in\mathbb{R}$ $\Large \Leftrightarrow \left\{\begin{align} & y > 0\\ & {\Delta}'=9-2y^2 < 0 \end{align}\right.$ $\Large \Leftrightarrow y > \dfrac{3\sqrt{2}}{2}$.

(1) $\Large \Leftrightarrow \log_{\sqrt{3}}\left(3(3x^2+1)\right)\leq \log_{\sqrt{3}}\left(yx^2-6x+2y\right)$

$\Large \Leftrightarrow 3(3x^2+1)\leq yx^2-6x+2y$ $\Large \Leftrightarrow (y-9)x^2-6x+2y-3\geq 0$, $\Large \forall x\in\mathbb{R}$

$\Large \Leftrightarrow \left\{\begin{align} & a=0\\ & bx+c\geq 0\end{align}\right.$ $\Large \cup$ $\Large \left\{\begin{align} & a > 0\\ & {\Delta}'\leq 0\end{align}\right.$ $\Large \Leftrightarrow \left\{\begin{align} & y=9\\ & -6x+15\geq 0\ \forall x \in (\text{Loai})\end{align}\right.$ $\Large \cup$ $\Large \left\{\begin{align} & y > 9\\ & 9-(y-9)(2y-3)\leq 0 \end{align}\right.$ $\Large \Leftrightarrow \left\{\begin{align} & y > 9\\ & -2y^2+21y-18\leq 0\end{align}\right.$ $\Large \Leftrightarrow y\geq \dfrac{21+3\sqrt{33}}{4}$

Do $\Large \left\{\begin{align} & y\in (-20; 20)\\ & y\in\mathbb{Z}\\ & y > \dfrac{3\sqrt{2}}{2}\\ & y\geq \dfrac{21+3\sqrt{33}}{4}\end{align}\right.$ $\Large \Rightarrow y\in \begin{Bmatrix}
10; 11;...; 18; 19
\end{Bmatrix}$

Vậy có 10 số nguyên $\Large y$ thỏa yêu cầu bài toán.