The values of k for which the inequality kcos2⁡x−kcos⁡x+1≥0,∀x∈(−∞,∞) holds is

kcos2⁡x−kcos⁡x+1≥0∀x∈(−∞,∞)⇒ kcos2⁡x−cos⁡x+1≥0      →ibut cos2⁡x−cos⁡x=cos⁡x−122−14⇒−14≤cos2⁡x−cos⁡x≤2Now, from (i), we get  2k+1≥0⇒k≥−12 Also, −k4+1≥0⇒ k≤4⇒ −12≤k≤4