The solution of inequality cos 2 x ≤ cos x is

Question

The solution of inequality cos 2 x ≤ cos x is (A) x ∈[2 n π-(π/3), 2 n π+(π/3)], n ∈ I (B) x ∈[2 n π-(2 π/3), 2 n π+(2 π/3)], n ∈ I (C) x ∈[2 n π, 2 n π+(2 π/3)], n ∈ I (D) x ∈[2 n π-(2 π/3), 2 n π], n ∈ I

Tardigrade Admin · Accepted Answer

cos 2 x ≤ cos x ⇒ 2 cos 2 x- cos x-1 ≤ 0 ⇒ 2 cos 2 x-2 cos x+ cos x-1 ≤ 0 ⇒ 2 cos x( cos x-1)+1( cos x-1) ≤ 0 ⇒( cos x-1)(2 cos x+1) ≤ 0 ⇒ cos x ∈[-(1/2), 1] ∴ x ∈[-(2 π/3), (2 π/3)] General soltution is x ∈[2 n π-(2 π/3), 2 n π+(2 π/3)], n ∈ I

Q. The solution of inequality $cos 2 x \leq cos x$ is

$x \in [2 nπ - \frac{π}{3}, 2 nπ + \frac{π}{3}], n \in I$

$x \in [2 nπ - \frac{2 π}{3}, 2 nπ + \frac{2 π}{3}], n \in I$

$x \in [2 nπ, 2 nπ + \frac{2 π}{3}], n \in I$

$x \in [2 nπ - \frac{2 π}{3}, 2 nπ], n \in I$

Q. The solution of inequality cos2x≤cosx is

x∈[2nπ−3π​,2nπ+3π​],n∈I

x∈[2nπ−32π​,2nπ+32π​],n∈I

x∈[2nπ,2nπ+32π​],n∈I

x∈[2nπ−32π​,2nπ],n∈I

cos2x≤cosx⇒2cos2x−cosx−1≤0 ⇒2cos2x−2cosx+cosx−1≤0 ⇒2cosx(cosx−1)+1(cosx−1)≤0 ⇒(cosx−1)(2cosx+1)≤0 ⇒cosx∈[−21​,1] ∴x∈[−32π​,32π​] General soltution is x∈[2nπ−32π​,2nπ+32π​],n∈I

Q. The solution of inequality $cos 2 x \leq cos x$ is

$x \in [2 nπ - \frac{π}{3}, 2 nπ + \frac{π}{3}], n \in I$

$x \in [2 nπ - \frac{2 π}{3}, 2 nπ + \frac{2 π}{3}], n \in I$

$x \in [2 nπ, 2 nπ + \frac{2 π}{3}], n \in I$

$x \in [2 nπ - \frac{2 π}{3}, 2 nπ], n \in I$