Given:

f(x) = 3x+cos3x

f’(x) = 3-3sin3x

f’(x) = 3(1-sin3x)

sin3x varies from[-1,1]

when sin3x is 1 f’(x) = 0 and sin3x is -1 f’(x) = 6

As the function is increasing in 0 to 6.

∴ The function is increasing on R.