Giải phương trình f ′ ( x ) = 0 .

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Cho hàm số \(f\left( x \right) = \frac{{{\rm{sin}}3x}}{3} - {\rm{cos}}x - \sqrt 3 \left( {{\rm{sin}}x - \frac{{{\rm{cos}}3x}}{3}} \right)\). Giải phương trình \(f'\left( x \right) = 0\).

\(x = \frac{\pi }{{12}} + k\pi \) hoặc \(x = - \frac{{3\pi }}{8} + k\frac{\pi }{2}\left( {k \in \mathbb{Z}} \right)\).

\(x = - \frac{\pi }{{12}} + k\pi \) hoặc \(x = - \frac{{3\pi }}{8} + k\frac{\pi }{2}\left( {k \in \mathbb{Z}} \right)\).

\(x = - \frac{\pi }{{12}} - k\pi \) hoặc \(x = \frac{{3\pi }}{8} + k\frac{\pi }{2}\left( {k \in \mathbb{Z}} \right)\).

\(x = \frac{\pi }{{12}} + k\pi \) hoặc \(x = \frac{{3\pi }}{8} + k\frac{\pi }{2}\left( {k \in \mathbb{Z}} \right)\).

Giải thích

Đáp án C

Hướng dẫn giải

f'x=(sin3x)'3−(cosx)'−3(sinx)'−(cos3x)'3

\( \Leftrightarrow f'\left( x \right) = \frac{{3{\rm{cos}}3x}}{3} + {\rm{sin}}x - \sqrt 3 \left( {{\rm{cos}} + \frac{{3{\rm{sin}}3x}}{3}} \right)\)

\( \Leftrightarrow f'\left( x \right) = {\rm{cos}}3x + {\rm{sin}}x - \sqrt 3 {\rm{cos}}x - \sqrt 3 {\rm{sin}}3x\)

\(f'\left( x \right) = 0\)

\( \Leftrightarrow {\rm{sin}}x - \sqrt 3 {\rm{cos}}x = - {\rm{cos}}3x + \sqrt 3 {\rm{sin}}3x\)

\( \Leftrightarrow \frac{1}{2}{\rm{sin}}x - \frac{{\sqrt 3 }}{2}{\rm{cos}}x = - \frac{1}{2}{\rm{cos}}3x + \frac{{\sqrt 3 }}{2}{\rm{sin}}3x\)

\( \Leftrightarrow {\rm{sin}}\left( {x - \frac{\pi }{3}} \right) = {\rm{sin}}\left( {3x - \frac{\pi }{6}} \right)\)

\( \Leftrightarrow x - \frac{\pi }{3} = 3x - \frac{\pi }{6} + k2\pi \) hoặc \(x - \frac{\pi }{3} = \pi - 3x + \frac{\pi }{6} + k2\pi \)

\( \Leftrightarrow x = - \frac{\pi }{{12}} - k\pi \) hoặc \(x = \frac{{3\pi }}{8} + \frac{{k\pi }}{2}\).