Find ‘a’ for which f(x) = a (x + sinx) + a is increasing on R.
f(x) = a (x + Sin x) + a
f’(x) = a (1 + Cos x) + 0
f’(x) = a (1 + Cos x)
For f(x), to be increasing, it must have,
f’(x) > 0
a (1 + Cos x) > 0 -------------- (i)
-1 ≤ Cos x ≤ 1, ∀x ϵ R
0 ≤ (1 + Cos x) ≤ 2, ∀x ϵ R
a > 0 {From eq. (i)}
a ϵ (0, ∞)
Hence the required set of values is a ϵ (0, ∞).