Question

Part of the domain of the function$f\left(x\right)=\sqrt{\frac{\cos x-\left({\displaystyle \frac{1}{2}}\right)}{6+35x-6x^2}}$ lying in the interval $\left[-1,6\right]$is

• A. $\left[\frac{-1}{6},\frac{\pi }{3}\right]$
• B. $\left[\frac{5\pi }{3},6\right]$
• C. $\left[\frac{-1}{5},\frac{\pi }{3}\right]\cup \left[\frac{5\pi }{3},6\right]$
• D. $(\frac{-1}{6},\frac{\pi }{3}]\cup \left[\frac{5\pi }{3},6\right]$

Answer 1

The correct option is D

$(\frac{-1}{6},\frac{\pi }{3}]\cup \left[\frac{5\pi }{3},6\right]$

Explanation for the correct option:

Finding the domain of the given function:

Given $f\left(x\right)=\sqrt{\frac{\cos x-\left({\displaystyle \frac{1}{2}}\right)}{6+35x-6x^2}}$

Simplifying the function

$$\begin{gathered} \cos x\ge \left(\frac{1}{2}\right),(6–x)(1+6x)>0\mathrm{or}\cos x\le \left(\frac{1}{2}\right),(6–x)(1+6x)<0 \\ x\in (-\frac{1}{6},\frac{\pi }{3}]\cup \left(\frac{5\pi }{3},6\right) \end{gathered}$$

Therefore, the correct answer is option (D).