Demonstrate that f(x) is an increasing function if f(x) = x + cosx .

To prove that a function is an increasing/decreasing one, we have to calculate the first derivative of that function.

f'(x)=1-sinx

Based on the fact that the function sin x < 1 and the maximum value is sin x =1 when x = pi/2, the difference: 1 - sin x is positive or, at much 0.

Because of the fact that the first derivative is positive, the function is increasing.