
MATHEMATICS  THEORY



In this section, the concept of a monotonic function is discussed and the method to find a function's monotonicity is introduced. 





Monotonicity

Increasing Function
Decreasing Function 

In engineering research, sometimes a diagram
can help the researcher better understand a function. A function's increasing
or decreasing tendency is useful when sketching a draft.
A function is called increasing on an interval if the function
value increases as the independent value increases. That is if x_{1} > x_{2},
then f(x_{1}) > f(x_{2}). On the other hand, a function
is called decreasing on an interval if the function value decreases as
the independent value increases. That is if x_{1} > x_{2},
then f(x_{1}) < f(x_{2}). A function's increasing
or decreasing tendency is called monotonicity on its domain. 



Example of Monotonic Function 

The monotonicity concept can be better understood
by finding the increasing and decreasing interval of the function, say
y = (x1)^{2}.
In the interval of (∞,
1], the function is decreasing. In the interval of [1, +∞), the
function is increasing. However, the function is not monotonic in its
domain (∞, +∞). 



Derivative and Monotonic


In the Derivative and Monotonic graphic
on the left, the function is decreasing in [x_{1}, x_{2}]
and [x_{3}, x_{4}],
and the slope of the function's tangent lines are negative. On the other
hand, the function is increasing in [x_{2}, x_{3}] and
the slope of the function's tangent line is positive. Is there any certain
relationship between monotonicity and derivative? The answer is yes and
is discussed below. 




Test for Monotonic Functions



Test for monotonic functions states:
Suppose a function is continuous on [a, b] and it is differentiable
on (a, b).

If the derivative is larger than zero for all x in (a, b), then the function is increasing on [a, b].

If the derivative is less than zero for all x in (a, b), then the function is decreasing on [a, b].




Function y = x^{2}  4 

The test for monotonic functions can be better
understood by finding the increasing and decreasing range for the function
f(x) = x^{2}  4.
The function f(x) = x^{2}  4 is a polynomial function, it is
continuous and differentiable in its domain (∞, +∞), and
thus it satisfies the condition of monatomic function test. In order
to find its monotonicity, the derivative of the function
needs to be calculated. That is
df(x)/dx = d(x^{2}  4)/dx
=
(d(x^{2})
 d(4))/dx
=
2x
It is obvious that the function df(x)/dx = 2x is negative when x < 0,
and it is positive when x > 0. Therefore, function f(x) = x^{2} 
4 is increasing in the range of (∞, 0) and decreasing in the range
of (0, +∞). This result is confirmed by the diagram on the left. 



