Solveeit Logo
Login

Question

Question: Part of the domain of the function \(f(x) = \sqrt{\frac{\cos x - \frac{1}{2}}{6 + 35 - 6x^{2}}}\) ly...

Part of the domain of the function f(x)=cos⁡x−126+35−6x2f(x) = \sqrt{\frac{\cos x - \frac{1}{2}}{6 + 35 - 6x^{2}}} lying in the interval [−1,6]\lbrack - 1,6\rbrack is

A

[−16,π3]∪[5π3,6]\left\lbrack - \frac{1}{6},\frac{\pi}{3} \right\rbrack \cup \left\lbrack \frac{5\pi}{3},6 \right\rbrack

B

[−16,π3]∪[5π3,6]\left\lbrack - \frac{1}{6},\frac{\pi}{3} \right\rbrack \cup \left\lbrack \frac{5\pi}{3},6 \right\rbrack

C

(−16,6)\left( - \frac{1}{6},6 \right)

D

None of these

Answer

[−16,π3]∪[5π3,6]\left\lbrack - \frac{1}{6},\frac{\pi}{3} \right\rbrack \cup \left\lbrack \frac{5\pi}{3},6 \right\rbrack

Explanation

Solution

The function f is meaningful only if cos⁡x−12≥0,6+35x−6x2>0\cos x - \frac{1}{2} \geq 0,6 + 35x - 6x_{2}^{> 0} or cos⁡x−12≤0,\cos x - \frac{1}{2} \leq 0, 6+35x−6x2<06 + 35x - 6x^{2} < 0 i.e. cos⁡x≥12,(6−x)(1+6x)>0\cos x \geq \frac{1}{2},(6 - x)(1 + 6x) > 0or

cos⁡x≤12,(6−x)(1+6x)<)\cos x \leq \frac{1}{2},(6 - x)(1 + 6x) < )

These inequalities are satisfied if x∈[−16,π3]∪[5π3,6]x \in \left\lbrack - \frac{1}{6},\frac{\pi}{3} \right\rbrack \cup \left\lbrack \frac{5\pi}{3},6 \right\rbrack.