First slide

Trigonometric equations

Question

If $\cos^{2} x - (c - 1) \cos x + 2 c \geq 6 for every x \in R$ then the true set of values of c is

Moderate

A

$[2, \infty)$
B

$[4, \infty)$
C

$(- \infty, - 2]$
D

$(- \infty, - 4]$

Solution

$\begin{array}{l} \cos^{2} x - (c - 1) \cos x + 2 c \geq 6 \forall x \in R \\ \Rightarrow (\cos x - 2) (\cos x - (c - 3)) \geq 0, \forall x \in R \\ \Rightarrow \cos x \leq c - 3 \\ \Rightarrow c - 3 \geq 1 \\ \Rightarrow c \in [4, \infty) \end{array}$

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