Let the range of the function f(x)=6+16 x (()/(3)-x) (()/(3)+x) 3 x 6 x, x R be [, ].

Question

Let the range of the function $f(x)=6+16 \cos x \cdot \cos \left(\frac{\pi}{3}-x\right) \cdot \cos \left(\frac{\pi}{3}+x\right) \cdot \sin 3 x \cdot \cos 6 x, x \in \mathbf{R}$ be $[\alpha, \beta]$. Then the distance of the point $(\alpha, \beta)$ from the line $3 x+4 y+12=0$ is :

Accepted Answer

$$\begin{aligned}
&\begin{aligned}
f(x) & =6+16\left(\frac{1}{4} \cos 3 x\right) \sin 3 x \cdot \cos 6 x \
& =6+4 \cos 3 x \sin 3 x \cos 6 x \
& =6+\sin 12 x
\end{aligned}\
\&\text { Range of } f(x) \text { is [5, 7] }\
\&\begin{aligned}
& (\alpha, \beta) \equiv(5,7) \
& \text { distance }=\left|\frac{15+28+12}{5}\right|=11
\end{aligned}
\end{aligned}$$

Solutions · 1