Tìm max min của y=1/2sin2x+cosx mọi ng giúp e với ạ 04/08/2021 Bởi Abigail Tìm max min của y=1/2sin2x+cosx mọi ng giúp e với ạ
Đáp án: $\begin{cases}miny = -\dfrac{3\sqrt3}{4} \Leftrightarrow x = -\dfrac{5\pi}{6} + k2\pi\\maxy = \dfrac{3\sqrt3}{4} \Leftrightarrow x = \dfrac{\pi}{6} + k2\pi \quad\end{cases}\quad (k \in \Bbb Z)$ Giải thích các bước giải: $y = \dfrac{1}{2}\sin2x + \cos x$ $= \sin x \cos x + \cos x$ Ta có: $9 – 4\sqrt3y = 9 – 4\sqrt3(\sin x \cos x + \cos x)$ $= 9(\sin^2x + \cos^2x) – 4\sqrt3\sin x\cos x – 4\sqrt3\cos x$ $= 2(\cos^2x – 2\sqrt3\sin x\cos x + 3\cos^2x) + (4\cos^2x – 4\sqrt3\cos x + 3)$ $= 2(\cos x – \sqrt3\sin x)^2 + (2\cos x – \sqrt3)^2 \geq 0$ $\Leftrightarrow 4\sqrt3y \leq 9$ $\Leftrightarrow y \leq \dfrac{3\sqrt3}{4}$ Dấu = xảy ra $\begin{cases}\cos x – \sqrt3\sin = 0\\2\cos x – \sqrt3 = 0\end{cases} \Leftrightarrow x = \dfrac{\pi}{6} + k2\pi \quad (k \in \Bbb Z)$ Tương tự, ta được: $9 + 4\sqrt3y = 9 + 4\sqrt3(\sin x \cos x + \cos x)$ $= 9(\sin^2x + \cos^2x) + 4\sqrt3\sin x\cos x + 4\sqrt3\cos x$ $= 2(\cos^2x + 2\sqrt3\sin x\cos x + 3\cos^2x) + (4\cos^2x + 4\sqrt3\cos x + 3)$ $= 2(\cos x + \sqrt3\sin x)^2 + (2\cos x + \sqrt3)^2 \geq 0$ $\Leftrightarrow 4\sqrt3y \geq -9$ $\Leftrightarrow y \geq -\dfrac{3\sqrt3}{4}$ Dấu = xảy ra $\begin{cases}\cos x + \sqrt3\sin = 0\\2\cos x + \sqrt3 = 0\end{cases} \Leftrightarrow x = -\dfrac{5\pi}{6} + k2\pi \quad (k \in \Bbb Z)$ Bình luận
Đáp án:
$\begin{cases}miny = -\dfrac{3\sqrt3}{4} \Leftrightarrow x = -\dfrac{5\pi}{6} + k2\pi\\maxy = \dfrac{3\sqrt3}{4} \Leftrightarrow x = \dfrac{\pi}{6} + k2\pi \quad\end{cases}\quad (k \in \Bbb Z)$
Giải thích các bước giải:
$y = \dfrac{1}{2}\sin2x + \cos x$
$= \sin x \cos x + \cos x$
Ta có:
$9 – 4\sqrt3y = 9 – 4\sqrt3(\sin x \cos x + \cos x)$
$= 9(\sin^2x + \cos^2x) – 4\sqrt3\sin x\cos x – 4\sqrt3\cos x$
$= 2(\cos^2x – 2\sqrt3\sin x\cos x + 3\cos^2x) + (4\cos^2x – 4\sqrt3\cos x + 3)$
$= 2(\cos x – \sqrt3\sin x)^2 + (2\cos x – \sqrt3)^2 \geq 0$
$\Leftrightarrow 4\sqrt3y \leq 9$
$\Leftrightarrow y \leq \dfrac{3\sqrt3}{4}$
Dấu = xảy ra $\begin{cases}\cos x – \sqrt3\sin = 0\\2\cos x – \sqrt3 = 0\end{cases} \Leftrightarrow x = \dfrac{\pi}{6} + k2\pi \quad (k \in \Bbb Z)$
Tương tự, ta được:
$9 + 4\sqrt3y = 9 + 4\sqrt3(\sin x \cos x + \cos x)$
$= 9(\sin^2x + \cos^2x) + 4\sqrt3\sin x\cos x + 4\sqrt3\cos x$
$= 2(\cos^2x + 2\sqrt3\sin x\cos x + 3\cos^2x) + (4\cos^2x + 4\sqrt3\cos x + 3)$
$= 2(\cos x + \sqrt3\sin x)^2 + (2\cos x + \sqrt3)^2 \geq 0$
$\Leftrightarrow 4\sqrt3y \geq -9$
$\Leftrightarrow y \geq -\dfrac{3\sqrt3}{4}$
Dấu = xảy ra $\begin{cases}\cos x + \sqrt3\sin = 0\\2\cos x + \sqrt3 = 0\end{cases} \Leftrightarrow x = -\dfrac{5\pi}{6} + k2\pi \quad (k \in \Bbb Z)$