we are given

(a)
Firstly, we will find critical numbers
so, we will find derivative

now, we can set it to 0
and then we can solve for x
we get

now, we can draw a number line and then locate these values
and then we can find sign of derivative on each intervals
increasing intervals:
![[0,\frac{3\pi}{4} )U(\frac{7\pi}{4} , 2\pi]](https://tex.z-dn.net/?f=%5B0%2C%5Cfrac%7B3%5Cpi%7D%7B4%7D%20%29U%28%5Cfrac%7B7%5Cpi%7D%7B4%7D%20%2C%202%5Cpi%5D)
Decreasing interval:

(b)
Local maxima:
It is the value of x where function changes from increasing to decreasing
so, local maxima is at

Local minima:
It is the value of x where function changes from decreasing to increasing
so, local minima is at

now, we will plug critical numbers and end values into original function
and we get
At x=0:


At
:


At
:


At
:


Global maxima:
It is the largest value among them
so, we get

Global minima:
It is the largest value among them
so, we get

(c)
now, we can find second derivative





now, we can set it to 0
and then we can solve for x

so, we get

now, we can draw number line and locate these values
and then we can find sign of second derivative on each intervals
concave up intervals:
![[0,\frac{\pi}{2})U(\frac{3\pi}{2}, 2\pi]](https://tex.z-dn.net/?f=%5B0%2C%5Cfrac%7B%5Cpi%7D%7B2%7D%29U%28%5Cfrac%7B3%5Cpi%7D%7B2%7D%2C%202%5Cpi%5D)
Concave down intervals:

Turning points:
All values of x for which concavity changes
so, we get turning points at
