Tìm GTNN,GTLN : sin3X -3sin²X

Đáp án:

$\begin{array}{l}
y = \sin 3x - 3{\sin ^2}x\\
 = 3\sin x - 4{\sin ^3}x - 3{\sin ^2}x\\
 =  - 4{\sin ^3}x - 3{\sin ^2}x + 3\sin x\\
Dat:{\mathop{\rm sinx}\nolimits}  = a\left( { - 1 \le a \le 1} \right)\\
 \Rightarrow y =  - 4{a^3} - 3{a^2} + 3a\\
 \Rightarrow y' =  - 12{a^2} - 6a + 3 = 0\\
 \Rightarrow  - 3.\left( {4{a^2} + 2a - 1} \right) = 0\\
 \Rightarrow \left[ \begin{array}{l}
a = \dfrac{{ - 1 - \sqrt 5 }}{4}\\
a = \dfrac{{ - 1 + \sqrt 5 }}{4}
\end{array} \right.\left( {tmdk} \right)\\
 \Rightarrow \left\{ \begin{array}{l}
{y_{ - 1}} =  - 4{a^3} - 3{a^2} + 3a =  - 2\\
{y_{\dfrac{{ - 1 - \sqrt 5 }}{4}}} = \dfrac{{ - 7 - 5\sqrt 5 }}{8}\\
{y_{\dfrac{{ - 1 + \sqrt 5 }}{4}}} = \dfrac{{ - 7 + 5\sqrt 5 }}{8}\\
{y_1} =  - 4
\end{array} \right.\\
 \Rightarrow \left\{ \begin{array}{l}
GTNN: =  - 4\\
GTLN:y = \dfrac{{ - 7 + 5\sqrt 5 }}{8}
\end{array} \right.
\end{array}$