Question

Tìm GTLN, GTNN

\(y=2cos3x+5sin\frac{3x}{2}-1\)

Answer 1

Lời giải:

$y=2(\cos ^2\frac{3x}{2}-\sin ^2\frac{3x}{2})+5\sin \frac{3x}{2}-1$

$=2(1-2\sin ^2\frac{3x}{2})+5\sin \frac{3x}{2}-1$

$=-4\sin ^2\frac{3x}{2}+5\sin \frac{3x}{2}+1$

Đặt $\sin \frac{3x}{2}=t$ ($t\in [-1;1]$) thì:

$y=-4t^2+5t+1$

$y'=-8t+5=0\Leftrightarrow t=\frac{5}{8}$

$y(1)=2; y(\frac{5}{8})=\frac{41}{16}; y(-1)=-8$

Vậy $y_{\max}=\frac{41}{16}; y_{\min}=-8$