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$1.$ $f(x) = {\mathop{\rm s}\nolimits} {\rm{inx}} +
\frac{1}{3}\sin 3x + \frac{2}{5}\sin 5x$
$
f'(x) = \cos x + \cos 3x + 2\cos 5x\\
f'(x) = 0 \Leftrightarrow 2\cos 3x\cos 2x + 2\cos 4x\cos x = 0\\
\Leftrightarrow (4\cos^3x - 3\cos x) \cos 2x + \cos 4x.\cos x = 0\\
\Leftrightarrow \cos x( ( 2\cos 2x - 1)\cos 2x + 2\cos^22x - 1) = 0 $
Đáp số: $\left[ \begin{array}{l}
x = \frac{\pi }{2} + k\pi \,\,(k \in Z)\\
x = \pm \frac{\alpha }{2} + k\pi \,(k \in Z,\,\,c{\rm{os}}\alpha =
\frac{{1 + \sqrt {17} }}{8})\\
x = \pm \frac{\beta }{2} + k\pi \,\,(k \in Z\,;\,c{\rm{os}}\beta =
\frac{{1\sqrt {17} }}{8})
\end{array} \right.$
$2.$ Dễ chứng minh
$\tan\frac{A}{2} \tan\frac{B}{2}+ \tan\frac{B}{2} \tan\frac{C}{2}
+\tan\frac{C}{2} \tan\frac{A}{2} =1$
nên đpcm $\Leftrightarrow \cos\frac{C}{2} (\sin\frac{A}{2} \cos\frac{B}{2}
+\sin\frac{B}{2} \cos\frac{A}{2} )+\sin\frac{C}{2} (\cos\frac{A}{2}
\cos\frac{B}{2} -\sin\frac{A}{2}\sin\frac{B}{2} )=1$
$\Leftrightarrow \cos\frac{C}{2} .\sin\frac{A+B}{2} +\sin\frac{C}{2}
.\cos\frac{A+B}{2} =1$
$\Leftrightarrow \cos^2\frac{C}{2} +\sin^2\frac{C}{2} =1 $ (đúng)