Question

How do you find the points where the graph of the function `f(x)=x+cosx` has horizontal tangents?

Answer 1

If a point on the graph of `f(x)` has a horizontal tangent line, then `f'(x) = 0` at the x-value corresponding to that point.

Hence

`f'(x)=1-sinx`

and for `f'(x)=0=>sinx=1=>x=2*k*pi+pi/2`

where `k` is an integer.

So at points `(2*k*pi+pi/2,f(2*k*pi+pi/2))` the graph of f(x) has horizontal tangents.