Giải phương trình \(f'\left( x \right) = 0,\) biết rằng
LG a
\(f\left( x \right) = 3x + {{60} \over x} - {{64} \over {{x^3}}} + 5\)
Lời giải chi tiết:
\(\begin{array}{l}
f'\left( x \right) = 3 - \dfrac{{60}}{{{x^2}}} - \dfrac{{64.\left( { - 3{x^2}} \right)}}{{{x^6}}}\\
= 3 - \dfrac{{60}}{{{x^2}}} + \dfrac{{192}}{{{x^4}}}\\
= \dfrac{{3{x^4} - 60{x^2} + 192}}{{{x^4}}}\\
f'\left( x \right) = 0\\
\Leftrightarrow \dfrac{{3{x^4} - 60{x^2} + 192}}{{{x^4}}} = 0\\
\Leftrightarrow 3{x^4} - 60{x^2} + 192 = 0\\
\Leftrightarrow \left[ \begin{array}{l}
{x^2} = 16\\
{x^2} = 4
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = \pm 4\\
x = \pm 2
\end{array} \right.
\end{array}\)
Vậy \(x\in\left\{ { \pm 2; \pm 4} \right\}.\)
LG b
\(\displaystyle f\left( x \right) = {{\sin 3x} \over 3} + \cos x\) \(\displaystyle - \sqrt 3 \left( {\sin x + {{\cos 3x} \over 3}} \right).\)
Lời giải chi tiết:
\(\begin{array}{l}
f'\left( x \right)\\
= \frac{{3\cos 3x}}{3} - \sin x - \sqrt 3 \left( {\cos x + \frac{{ - 3\sin 3x}}{3}} \right)\\
= \cos 3x - \sin x - \sqrt 3 \left( {\cos x - \sin 3x} \right)\\
= \cos 3x + \sqrt 3 \sin 3x - \sin x - \sqrt 3 \cos x\\
f'\left( x \right) = 0\\
\Leftrightarrow \cos 3x + \sqrt 3 \sin 3x - \sin x - \sqrt 3 \cos x = 0\\
\Leftrightarrow \cos 3x + \sqrt 3 \sin 3x = \sin x + \sqrt 3 \cos x\\
\Leftrightarrow \frac{1}{2}\cos 3x + \frac{{\sqrt 3 }}{2}\sin 3x = \frac{1}{2}\sin x + \frac{{\sqrt 3 }}{2}\cos x\\
\Leftrightarrow \cos \left( {3x - \frac{\pi }{3}} \right) = \cos \left( {x - \frac{\pi }{6}} \right)\\
\Leftrightarrow \left[ \begin{array}{l}
3x - \frac{\pi }{3} = x - \frac{\pi }{6} + k2\pi \\
3x - \frac{\pi }{3} = - x + \frac{\pi }{6} + k2\pi
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
2x = \frac{\pi }{6} + k2\pi \\
4x = \frac{\pi }{2} + k2\pi
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
x = \frac{\pi }{{12}} + k\pi \\
x = \frac{\pi }{8} + \frac{{k\pi }}{2}
\end{array} \right.
\end{array}\)
