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Giải phương trình \(f’\left( x \right) = 0\) biết
a) \(f\left( x \right) = \sqrt 3 \cos x + \sin x – 2x – 5\)
b) \(f\left( x \right) = {{2\cos 17x} \over {17}} – {{\sqrt 3 \sin 5x} \over 5} + {{\cos 5x} \over 5} + 2\)
a) Với mọi \(x \in R\) ta có
\(\eqalign{& f’\left( x \right) = – \sqrt 3 \sin x + \cos x – 2 \cr& f’\left( x \right) = 0 \Leftrightarrow {1 \over 2}\cos x – {{\sqrt 3 } \over 2}\sin x = 1\cr& \Leftrightarrow \cos x.\cos {\pi \over 3} – \sin x.\sin {\pi \over 3} = 1 \cr& \Leftrightarrow \cos \left( {x + {\pi \over 3}} \right) = 1 \Leftrightarrow x + {\pi \over 3} = k2\pi \cr&\Leftrightarrow x = – {\pi \over 3} + k2\pi \,\,\left( {k \in Z} \right) \cr} \)
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b) Với mọi \(x \in R\) ta có
\(\eqalign{& f’\left( x \right) = – 2\sin 17x – \sqrt 3 \cos 5x – \sin 5x \cr& f’\left( x \right) = 0\cr& \Leftrightarrow \sin 17x + \left( {{{\sqrt 3 } \over 2}\cos 5x + {1 \over 2}\sin 5x} \right) = 0 \cr& \Leftrightarrow \sin 17x + \left( {\sin {\pi \over 3}\cos 5x + \cos {\pi \over 3}\sin 5x} \right) = 0 \cr& \Leftrightarrow \sin \left( {5x + {\pi \over 3}} \right) = \sin \left( { – 17x} \right) \cr} \)
\( \Leftrightarrow \left[ \matrix{5x + {\pi \over 3} = – 17x + k2\pi \hfill \cr5x + {\pi \over 3} = \pi + 17x + k2\pi \hfill \cr} \right. \Leftrightarrow \left[ \matrix{x = – {\pi \over {66}} + {{k\pi } \over {11}} \hfill \cr x = – {\pi \over {18}} – {{k\pi } \over 6} \hfill \cr} \right.\)