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. `a in (-5,5)`
B. `a in (-oo,-5)`
C. `a in (5,oo)`
D. none of these

Answer

B

Explanation

We have
` f(x)=ax+ 3 sin x + 4 cos x `
`f(x)=a+3 cos x- 4 sin x `
If f(x) is vertible then
`f(x) gt 0 for all or f (x) lt 0 for all x `
`rArr a + 3 cos x-4 sin x lt 0 for all x `
or `a+3 cos x-4 sin x lt 0 for all x `
`rArr a lt 5 or a +5 lt 0 " "[because -5 lt 3 cos x -4sin cx le 5 ]`
`rArr a gt or, lt -5 rArr a in (-oo ,-5) cup (5, oo)`