Let $f: R \rightarrow R$ be a function such that $f(x)=a x+3 \sin x+4 \cos x$. Then, f(x) is invertible if 

none of these

We have,

$f(x) =a x+3 \sin x+4 \cos x$

$\Rightarrow f'(x) =a+3 \cos x-4 \sin x$ 

If f(x) is invertible, then 

f'(x) > 0 for all x  or,  f'(x) < 0 for all x

$\Rightarrow a+3 \cos x-4 \sin x>0$ for all  x 

or, $a+3 \cos x-4 \sin x<0$ for all x 

$\Rightarrow a-5>0$ or, $a+5<0$               $[∵-5 \leq 3 \cos x-4 \sin x \leq 5]$ 

$\Rightarrow a>5$ or, $a<-5 \Rightarrow a \in(-\infty,-5) \cup(5, \infty)$