f(x)=x+cos x

To find if the function is decreasing or increasing we need to determine the derivative.

If the first derivative is positive, then the function is increasing. If the first derivative is negative, then the function is decreasing.

f'(x)= 1 -sinx

sinx < 1 then 1-sinx >0

f'(x) > 0 , then the function is increasing

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.

f(x) = x+cosx. To demonstrate this increasing for function:

Solution: We know that any function for which its derivative is postive is an incresing function.

f'(x) = d/dx(x+cosx) = 1-sinx which is postive for all x, as -1 sign x < 1 . So f(x) is an incresing function.