Monotone Function and Its Derivative

 

Motivation:

                          For a function f(x),  it is known that its derivative function is defined as

 

                                            

 

                         If the value of this limit is larger or smaller than 0,  what would the function look like?  That is the topic we are going to explore here.

 

 

 

 

 

Definition ( Monotone Function )

                   Let f(x) be a function of x.  f(x) is known as  increasing ( or monotonically increasing ) if

 

                                                   f(a)  f(b)  for any a < b  .

 

                  Similarly,  f(x) is known as  decreasing ( or monotonically decreasing ) if

 

                                                  f(a)  f(b)  for any  a< b .

 

                  If the “equality” is removed in the statement above, it is known as “strictly increasing or strictly decreasing”  function.

 

 

 

 

 

 

Theorem      If  f’(x) > 0  for x  [a,b] ,  then  f(x) is  monotonically

                      increasing in the interval [a,b]  .

 

    Proof:

                         Let d  [a, b]  and  f’(d) = c > 0 . 

                        So, for any  >0 ,  there exists   > 0 such that

 

                                         <   when  | h| <

                     c -  <  < c +      when |h| <

                      Since c>0,  we can choose  small enough such that  c-  > 0.

 

                      So,   if h > 0,

                                           0 < h (c -  ) <   f(d+h) – f(d)

                      

                              if h < 0,

                                               f(d+h) –f (d) <  h ( c -  )  < 0

 

                    Then in the small neighborhood of  d ,  f(x) is monotonically increasing. And this holds for every point in the interval [a,b] because  f’(x)  > 0 in this interval.  So, f(x) is monotonically increasing in this interval.

 

 

 

Example:  f(x)=x² – 2x-3

 

                   f’(x) = 2x -2

                  Thus,   when x > 1,  f’(x) > 0 ; when x< 1,  f’(x) < 0  .  Roughly speaking, f(x) should look like as follows:

 

                                       

                 While x< 1,  f(x) is decreasing ;  while x>1,  f(x) is monotonically increasing.

 

 

 

 

 

 

Example:   f(x)=x³ -2x² -4x+1 

 

                    f’(x) = 3x² – 4x -4 = (3x+2)(x-2)

 

                                             

                  When x < - ,  f(x)  is monotonically increasing because  f’(x) > 0 .

                  When   -  < x < 2 ,   f(x) is monotonically decreasing because f’(x) < 0;

                  When   x > 2,  f(x) is monotonically increasing because  f’(x) > 0.

 

 

 

 

 

 

Example:     f(x) = ln(x) where x > 0  .

 

                       f’(x) =  > 0 for x > 0 .

 

                    Thus,  f(x) is monotonically increasing in its domain.