Question

Critical numbers:
g(x) = x+2sinx 0<=x<=2pi
Textbook explains that since g is differentiable everywhere, then the only critical numbers are 2pi/3 and 4pi/3.

Answer 1

Why is g differentiable everywhere? Shouldn't it be diff'ble only at 0<=x<=2pi?

Answer 2

the text books wants the value of the critical point to lie between 0 and 2pi. However, it is true that the function is differenciable everywhere. As a matter of fact, there are infinitely many critical points

Answer 3

So g is diff'ble everywhere BUT I only need to worry about the critical values between 0 and 2pi.
I was having trouble understanding why it's diff'ble everywhere, but I understand better from the graph that it's because sin and cos oscillate infinitely.