Increasing and Decreasing functions

2 min read

Increasing - Non-decreasing functions Decreasing - Non-increasing functions

The first derivative of a function is used to check for increasing and decreasing functions.

Definition:

  • Increasing Function - A function f(x) is said to be increasing on an interval I if for any two numbers x and y in I such that x < y, we have f(x) ≤ f(y).
  • Decreasing Function - A function f(x) is said to be decreasing on an interval I if for any two numbers x and y in I such that x < y, we have f(x) ≥ f(y).
  • Strictly Increasing Function - A function f(x) is said to be strictly increasing on an interval I if for any two numbers x and y in I such that x < y, we have f(x) < f(y).
  • Strictly Decreasing Function - A function f(x) is said to be strictly decreasing on an interval I if for any two numbers x and y in I such that x < y, we have f(x) > f(y).

Graphical Representation:

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Suppose a function f(x) is differentiable on an open interval I, then we have

  • If f'(x) ≥ 0 on I, the function is said to be an increasing function on I.
  • If f'(x) ≤ 0 on I, the function is said to be a decreasing function on I.

Properties:

  • If the functions f and g are increasing functions on an open interval I, then the sum of the functions f + g is also increasing on this interval.
  • If the functions f and g are decreasing functions on an open interval I, then the sum of the functions f + g is also decreasing on this interval.
  • If the function f is an increasing function on an open interval I, then the opposite function -f is decreasing on this interval.
  • If the function f is a decreasing function on an open interval I, then the opposite function -f is increasing on this interval.
  • If the function f is an increasing function on an open interval I, then the inverse function 1/f is decreasing on this interval.
  • If the function f is a decreasing function on an open interval I, then the inverse function 1/f is increasing on this interval.
  • If the functions f and g are increasing functions on an open interval I and f, g ≥ 0 on I, then the product of the functions fg is also increasing on this interval.
  • If the functions f and g are decreasing functions on an open interval I and f, g ≥ 0 on I, then the product of the functions fg is also decreasing on this interval.
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