Question

Let $f: R \rightarrow R$ defined as $f(x)=\frac{x}{2}+\frac{\sin 2 x}{4}-(2-c) \sin x-2 c x$ is strictly increasing on $R$, then the largest integral value of $c$ is equal to

• A. 0
• B. 1
• C. -1
• D. -2

Answer 1

Answer: (C)

$\Theta f(x)=\frac{x}{2}+\frac{\sin 2 x}{4}-(2-c) \sin x-2 c x $
$f^{\prime}(x)=\frac{1}{2}+\frac{\cos 2 x}{2}-(2-c) \cos x-2 c $
$f ^{\prime}( x )=\cos ^2 x +( c -2) \cos x -2 c$
$\Rightarrow f ^{\prime}( x )=(\cos x + c )(\cos x -2) \geq 0$
$\Rightarrow \cos x+ c \leq 0 $
$\Rightarrow c+1 \leq 0 \Rightarrow c \leq-1$