Let `f: RvecR` be a function such that `f(x)=a x+3sinx+4cosxdot` Then `f(x)` is invertible if `a in (-5,5)` (b) `a in (-oo,5)` `a in (-5,+oo)` (d) none of these

Options

A. `klt1`
B. `kgt1`
C. `klt2`
D. `kgt2`

Answer

4

Explanation

f(X) =a+3 cos x -4 sinx
`=a+5 cos (x+alpha)` where `cos alpha =3/5`
for invertible f(X) must be monotonic .Thus
`f(x) ge0 forall x or f(X) leoforallx`
i.e `a + 5 cos (x+alpha)ge 0 or a+5 cos(x+alpha)le0`
i.e `age-5 cos(x+alpha) or a le5 cos (x+alpha)`
i.e `agt5 or ale-5`