f(x)=x + cos x. Prove that f(x) is an increasing function.

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We can demonstrate if a function is a increasing/decreasing one, using the first derivative of that function.

f'(x)=1-sinx

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.

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