How the maximum and minimum values of a function can be found?
We can find the maximum and minimum values of a function by using the first derivative y':
If the function " f " is:
Then, there are two cases:
Case 1
In this case, function f has a local maximum value at c, its coordinates (C, F(C))
Case 2
In this case, function f has a local minimum value at c, its coordinates (C, F(C))
Note:
The previous theory remains true if f'(x) does not exist. However; f(x) is connected at x=c
How to find the maximum / Minimum values of a function using the first derivative y':
 We find f ′ (x) (the derivative with respect to "x")
 We put f ′ (x) = 0 (we find the derivative roots)
 We study the derivative signal before and after its roots, and there are 4 cases:
The Case (around x=c) 
What does it mean 
The function f has a local minimum value, at x = c, its coordinates are (c, f (c)). 

The function f has a local maximum value at x = c , its coordinates are (c, f (c)). 

The function f has no maximum nor minimum values (The function is increasing on its domain) 

The function f has no maximum nor minimum values (The function is decreasing on its domain) 
Notes
 The monotonic function is a function which is either entirely increasing or decreasing on its domain.
 If f (x) is:
 a function defined for the period [a, b]
 has a local maximum or minimum value at x = c where c ∊ ] a, b [
Then, if f´ (c) is known then f´ (c) = 0  Sometimes f´ (c) may equal 0 (f´ (c)=0). However; the function f has no maximum nor minimum values at x = c (the function is monotonic)