f(x)=x + cos x. Prove that f(x) is an increasing function.
We can demonstrate if a function is a increasing/decreasing one, using the first derivative of that function.
We know that the function sin x < 1, or the maximum value is sin x =1 when x = pi/2.
So the difference: 1 - sin x >=0.
If the first derivative of the function is positive, then the function is increasing.