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
\(\therefore\) a (1 + Cos x) > 0 ---------- (i)
\(\because\) 1 ≤ Cos x ≤ 1, ∀ x ∈ R
\(\therefore\) 0 ≤ (1 + Cos x) ≤ 2,∀ x ∈ R
\(\therefore\) a > 0 {From eq. (i)}
\(\therefore\) a ∈ (0, ∞)
Hence the required set of values is a ∈ (0, ∞).
