Maxima minina on an interval (calculus+trig)

[quicksilver123]

I need to find the max/min of a function on an interval.
The function is f(x)=x+cos(x) and the interval is <-PI,2pi>
There is an attached solution but I do not understand how to arrive at the given solution (see screenshot). I would personally just take the derivative as
F'(x)=1-sin(x)
However the solution says this is only half the answer and I do not understand the reasoning (I am trying to do this algebraicly, without thinking of it graphically).

 

[haruspex]

the solution says this is only half the answer

I think you may be misinterpreting what it says. It is strangely worded. Better would be "f'(x)=1-sin(x), which is ≥0 since sin(x)≤1 for any real x".

 

[quicksilver123]

Thanks