18: Derivative is
f ′(x) = 1 + sin(x).
Critical points: f = 0 for
x = −π/2 + kπ;
there are no points in D( f ) where
f ′ does not exist.
Since
f ′(x) ≥ 0
is always true, f is non-decreasing on
ℝ. Moreover, with the
exception of the critical points above,
f ′(x) > 0, so in fact f is
increasing on ℝ.
There are no local extrema, we just have horizontal tangent lines at the
critical points.
Now determine concavity using f ′′.
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Answer