Toán Lớp 11: Tìm gtln , gtnn
y=sin ²x + 2sinxcosx – cos ²x +5
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Toán Lớp 11: Tìm gtln , gtnn
y=sin ²x + 2sinxcosx – cos ²x +5
Comments ( 2 )
y = {\sin ^2}x – {\cos ^2}x + 2\sin x\cos x + 5\\
= \sin 2x – \cos 2x + 5\\
= \sqrt 2 \sin \left( {2x – \dfrac{\pi }{4}} \right) + 5\\
– 1 \le \sin \left( {2x – \dfrac{\pi }{4}} \right) \le 1\\
\to – \sqrt 2 \le \sin \left( {2x – \dfrac{\pi }{4}} \right) \le \sqrt 2 \\
\Rightarrow 5 – \sqrt 2 \le t \le 5 + \sqrt 2 \\
\Rightarrow \left\{ \begin{array}{l}
\max y = 5 + \sqrt 2 \Rightarrow 2x – \dfrac{\pi }{4} = \dfrac{\pi }{2} + k2\pi \\
\min y = 5 – \sqrt 2 \Rightarrow 2x – \dfrac{\pi }{4} = – \dfrac{\pi }{2} + k2\pi
\end{array} \right.\\
\Rightarrow \left\{ \begin{array}{l}
\max y = 5 + \sqrt 2 \Rightarrow x = \dfrac{{3\pi }}{8} + k\pi \\
\min t = 5 – \sqrt 2 \Rightarrow x = – \dfrac{\pi }{8} + k\pi
\end{array} \right.\left( {k \in \mathbb{Z}} \right)
\end{array}$