Conditions for Increasing and Decreasing functions

                        
Let I be an open interval contained in the domain of a real valued function ’f’. Then ‘f’ is said to be
(i) Increasing on I if x₁ < x₂ in I  f(x₁) ≤ f(x₂) for all x₁, x₂ Є I        
(ii) Strictly increasing on I if x₁ < x₂ in I  f(x₁) < f(x₂) for all x₁, x₂ Є I
(iii) Decreasing on I if x₁ < x₂ in I  f(x₁) ≥ f(x₂) for all x₁, x₂ Є I
(iv) Strictly decreasing on I if x₁ < x₂ in I  f(x₁) > f(x₂) for all x₁, x₂ Є I

Dependence on Differentiability

Let ‘f’ be continuous on [a, b] and differentiable on the open interval (a, b).  Then f is increasing in [a, b] if f’(x) > 0 for each x Є (a, b)
(i) f is decreasing in [a, b] if f’(x) < 0 for each x Є (a, b).
(ii) f is a constant function in [a, b] if f’(x) =0  for each x Є (a, b).
Solved Examples:
1) Show that the function given by f(x) = 5x + 19 is strictly increasing on R
F(x) = 5x + 19
F’(x) = 5 > 0 for all x Є R
Thus f(x) is strictly increasing on R

2) Find the intervals in which the function f given by f(x) = x² - 4x + 6 is
a) Strictly increasing      
b) Strictly decreasing
F(x) = x² – 4x + 6                                                   
F’(x) = 2x - 4                                              -∞           2                +∞ 
F’(x) = 0 implies 2x – 4 = 0, x = 2
In the interval (-∞, 2), f’(x) = 2x – 4 < 0, so it is strictly decreasing in this interval.
In the interval (2, ∞), f’(x) > 0, so it is strictly increasing in this interval

Graphical representation of increasing and decreasing functions

             (Increasing function)              (Strictly increasing function)
        

        (Strictly Decreasing function)                     (Decreasing function)

Example: Where the given function is increasing or decreasing:
f (x) = x³ - 4x, for x in the interval [-1, 2]
Solution:      

Starting from -1 (the beginning of the interval [-1, 2]):
At x = -1 the function is decreasing, it continues to decrease until about 1.2, it then increases from there, past x = 2
Within the interval [-1, 2]:
The curve decreases in the interval [-1, approximately 1.2]
The curve increases in the interval [approximately 1.2, 2]

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Reference Links:

• http://en.wikipedia.org/wiki/Interval_(mathematics)
• http://en.wikipedia.org/wiki/Domain_(ring_theory)
• http://en.wikipedia.org/wiki/Continuous_function
• http://www.opensourcemath.org/books/calc1-sage/html/Increasing_decreasing_funct.html