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
Answer & explanation
Correct answer: option 4
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, $a<-5 \Rightarrow a \in(-\infty,-5) \cup(5, \infty)$