Toán tìm m để hàm số y=căn 5sin4x-6cos4x+2m-1 xác định với mọi x 04/10/2021 By Faith tìm m để hàm số y=căn 5sin4x-6cos4x+2m-1 xác định với mọi x
$$\eqalign{ & y = \sqrt {5\sin 4x – 6\cos 4x + 2m – 1} \cr & DKXD:\,\,\,5\sin 4x – 6\cos 4x + 2m – 1 \ge 0\,\,\forall x \cr & \Rightarrow {5 \over {\sqrt {61} }}\sin 4x – {6 \over {\sqrt {41} }}\cos 4x \ge {{1 – 2m} \over {\sqrt {61} }} \cr & Dat\,\,\left\{ \matrix{ {5 \over {\sqrt {61} }} = \cos \alpha \hfill \cr {6 \over {\sqrt {61} }} = \sin \alpha \hfill \cr} \right. \cr & \Rightarrow \sin 4x\cos \alpha – \cos 4x\sin \alpha \ge {{1 – 2m} \over {\sqrt {61} }}\,\,\forall x \cr & \Leftrightarrow \sin \left( {4x – \alpha } \right) \ge {{1 – 2m} \over {\sqrt {61} }}\,\,\forall x \cr & \Leftrightarrow – 1 \ge {{1 – 2m} \over {\sqrt {61} }} \cr & \Leftrightarrow – \sqrt {61} \ge 1 – 2m \cr & \Leftrightarrow – \sqrt {61} – 1 \ge – 2m \cr & \Leftrightarrow \sqrt {61} + 1 \le m \cr} $$ Trả lời
$$\eqalign{
& y = \sqrt {5\sin 4x – 6\cos 4x + 2m – 1} \cr
& DKXD:\,\,\,5\sin 4x – 6\cos 4x + 2m – 1 \ge 0\,\,\forall x \cr
& \Rightarrow {5 \over {\sqrt {61} }}\sin 4x – {6 \over {\sqrt {41} }}\cos 4x \ge {{1 – 2m} \over {\sqrt {61} }} \cr
& Dat\,\,\left\{ \matrix{
{5 \over {\sqrt {61} }} = \cos \alpha \hfill \cr
{6 \over {\sqrt {61} }} = \sin \alpha \hfill \cr} \right. \cr
& \Rightarrow \sin 4x\cos \alpha – \cos 4x\sin \alpha \ge {{1 – 2m} \over {\sqrt {61} }}\,\,\forall x \cr
& \Leftrightarrow \sin \left( {4x – \alpha } \right) \ge {{1 – 2m} \over {\sqrt {61} }}\,\,\forall x \cr
& \Leftrightarrow – 1 \ge {{1 – 2m} \over {\sqrt {61} }} \cr
& \Leftrightarrow – \sqrt {61} \ge 1 – 2m \cr
& \Leftrightarrow – \sqrt {61} – 1 \ge – 2m \cr
& \Leftrightarrow \sqrt {61} + 1 \le m \cr} $$