f(x) = x + Cos x + ax + b
f’(x) = 1 – Sin x + a + 0
f’(x) = 1 – Sin x + a
For, f(x) to be increasing, it must have
f’(x) > 0
\(\therefore\) 1 – Sin x + a > 0
1 > Sin x - a
Sin x < a + 1
\(\because\) the maximum value of Sin x is 1.
Also, 1 < a + 1
a > 0
\(\therefore\) a ∈ (0, ∞)
Hence the required set of values is a ∈ (0, ∞).