Increasing and Decreasing Functions

Increasing and Decreasing Functions:

  • Increasing Function: A function f(x) is increasing on an interval if, as x increases, the corresponding values of f(x) also increase.
  • Decreasing Function: A function f(x) is decreasing on an interval if, as x increases, the corresponding values of f(x) decrease.

Using Derivative Rules for Analysis:

  1. Increasing Functions:

    • For a function f(x):
      • f(x)>0 on an interval I implies f(x) is increasing on I.
    • Example: f(x)=x2:
      • Calculate f(x)=2x.
      • f(x)>0 for x>0, indicating f(x) is increasing on x>0.
  2. Decreasing Functions:

    • For a function f(x):
      • f(x)<0 on an interval I implies f(x) is decreasing on I.
    • Example: f(x)=x3:
      • Calculate f(x)=3x2.
      • f(x)<0 for x0, indicating f(x) is decreasing on x0.

Using Derivative Rules for Increasing/Decreasing Functions:

  1. Increasing Functions:

    • If f(x)>0 for all x in an interval, then f(x) is increasing on that interval.
    • If f(x) is non-negative (or f(x)0) for all x in an interval, then f(x) is increasing on that interval.
  2. Decreasing Functions:

    • If f(x)<0 for all x in an interval, then f(x) is decreasing on that interval.
    • If f(x) is non-positive (or f(x)0) for all x in an interval, then f(x) is decreasing on that interval.

Example:

Example 1: Determining Increasing/Decreasing Intervals

Given f(x)=x33x29x+5:

  1. Find where f(x) is increasing or decreasing.

Solution:

  1. Derivative Calculation:

    • Calculate f(x)=ddx(x33x29x+5)=3x26x9
  2. Identifying Increasing/Decreasing Intervals:

    • Setting f(x)>0 for increasing intervals: 3x26x9>0 x22x3>0 (x3)(x+1)>0
      • This inequality holds for x<1 or x>3, indicating f(x) is increasing on x<1 or x>3.
    • Setting f(x)<0 for decreasing intervals: 3x26x9<0 x22x3<0 (x3)(x+1)<0
      • This inequality holds for 1<x<3, indicating f(x) is decreasing on 1<x<3.

Strategies:

  • Derivative Sign Analysis: Understand the relationship between the sign of the derivative and the behavior of the function (increasing or decreasing) on specific intervals.
  • Interval Identification: Use the derivative sign to determine intervals where the function exhibits increasing or decreasing behavior.

Example 1: Determining Increasing/Decreasing Behavior

Given f(x)=x22x:

  1. Find where f(x) is increasing or decreasing.

Solution:

  1. Derivative Calculation:
    • Find f(x) to determine increasing/decreasing behavior: f(x)=ddx(x22x)=2x2
    • Setting f(x)=0 to find critical points: 2x2=0 x=1
    • Test intervals around the critical point x=1:
      • Test x=0 (left of x=1): f(0)=2(0)2=2(Decreasing)
      • Test x=2 (right of x=1): f(2)=2(2)2=2(Increasing)

Strategies:

  • Derivative Analysis: Use the derivative f(x) to determine the function's increasing or decreasing behavior.
  • Critical Points: Identify critical points by setting f(x) equal to zero or where it is undefined to find potential points of change in behavior.
  • Test Intervals: Test intervals around critical points by selecting test points and evaluating f(x) to determine the function's behavior within those intervals.