Consider the function defined on

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Consider the function

f(x)=x+\sin x

defined on

\mathbb R

. Answer the following: (i) Find

f'(x)

. (ii) Show that

f'(x)\ge 0

for all

x\in\mathbb R

and determine the points where

f'(x)=0

. (iii) Investigate whether the sign of

f'(x)

changes at any point where

f'(x)=0

. Hence, state with reasons whether

f

has any local maxima or minima on

\mathbb R

. (iv) Describe the monotonic nature of

f

\mathbb R