Answer:
Step-by-step explanation:
f(t) = t + cos(t)
Criticals :
f' (t) = 1 - sin (t) = 0
sin (t) = 1
Given that the interval is : [ - 2π , 2π ]
thus;  t = π/2 and -3π/2
The region then splits into [ -2π , -3π/2 ], [-3π/2 , 2π ]  and [ π/2 , 2π ]
Region 1:                            Region 2:                         Region 3:
[ -2π , -3π/2 ]                      [-3π/2 , 2π ]                      [ π/2 , 2π ]
Test value (t): -11 π/6          Test value(t) = 0              Test value = π
f'(t) = 1 - sin(t)                       f'(t) = 1 - sin(t)                   f'(t) = 1 - sin(t)
f'(t) = 1 - sin (-11 π/6)            f'(t) = 1 - sin (0)                  f'(t) = 1 - sin(π)
f'(t) =  positive value           f'(t) = 1 - 0                         f'(t) = 1 - 0.0548
thus; it is said to be            f'(t) = 1  (positive)              f'(t) = 0.9452 (positive)
increasing.                          so it is increasing            so it is increasing
So interval of increase is :  [ -2π , 2π ]
There is no  local maximum value or minimum value since the function increases monotonically over [ -2π , 2π ]. Hence, there is no change in the pattern.
c) Inflection Points;
Given that :
f'(t) = 1 - sin (t)
Then f''(t) = - cos (t) = 0
within  [ -2π , 2π ], there exists 4 values of  t for which costs = 0
These are:
[-3π/2 ]
[-π/2 ]
[π/2 ]
[3π/2 ]
For Concativity:
This splits the region into [ -2π , -3π/2], [ -3π/2 ,  -π/2], [-π/2 , π/2] , [π/2 , 3π/2] and [ 3π/2 , 2π].
Region 1: [ -2π , -3π/2]      Region 2: [ -3π/2 ,  -π/2]            
Test value = - 11π/6           Test value = π                             
f''(t) = - cos (t)                      f''(t) = - cos (t)
- cos (- 11π/6) = negative    f''(-π) = - cos (- π)
Thus; concave is down.      f''(π) = -cos (π)
                                            positive, thus concave is up
Region (3):   [-π/2 , π/2]
Concave is down
Region (4):  [π/2 , 3π/2] 
Concave is up
Region (5):  [ 3π/2 , 2π]
Concave is down
We conclude that:
Concave up are at region 2 and 4:  [ -3π/2 ,  -π/2] ,  [π/2 , 3π/2] 
Concave down are at region 1,3 and 5 :  [ -2π , -3π/2] , [-π/2 , π/2] , [ 3π/2 , 2π]