- Hướng dẫn giải

Đặt

$$\begin{gathered} \left\{\begin{array}{l}\sqrt[3]{m+3\sin \left(x\right)}=a\\ \sin \left(x\right)=b\end{array}\right. \\ \iff \left\{\begin{array}{l}m+3a=b^3\\ m+3b=a^3\end{array}\right. \end{gathered}$$

Trừ 2 vế ta được

$$\begin{gathered} a^3-b^3=3b-3a \\ \iff \left(a-b\right)\left(a^2+ab+b^2\right)+3\left(a-b\right) \\ \iff \left(a-b\right)\left(a^2+ab+b^2+3\right)=0 \end{gathered}$$

Do$\left(a^2+ab+b^2+3\right)>0\forall a,b$

$\Rightarrow a=b\iff \sqrt[3]{m+3\sin \left(x\right)}=\sin \left(x\right)$

$$\begin{gathered} \iff m+3\sin \left(x\right)={\sin }^3\left(x\right) \\ \iff m={\sin }^3\left(x\right) - 3\sin \left(x\right) \end{gathered}$$

Đặt sinx = t, $t\in \left[-1;1\right]$

$$\begin{gathered} f\left(t\right)=t^3-3t \\ \Rightarrow f\text{'}\left(t\right)=3t^{}-3\le 0\forall t\in \left[-1;1\right] \end{gathered}$$

Đáp án cần chọn là A