CourseCourseQ BankQuestion BankDocsDocuments

5.3 Determining Intervals on Which a Function Is Increasing or Decreasing

Syllabus
2020
Topic
5.3
Level
—

Learning objectives

—FUN-4.A—Justify conclusions about the behavior of a function based on the behavior of its derivatives• FUN-4.A Justify conclusions about the behavior of a function based on the behavior of its derivatives.• FUN-4.A.1 The first derivative of a function can provide information about the function and its graph, including intervals where the function is increasing or decreasing.• Enduring understanding FUN-4: A function’s derivative can be used to understand some behaviors of the function.

Read Increasing and Decreasing Behavior from $f'$

The sign of f′(x)f'(x) tells the direction of change of ff. On an interval where f′(x)>0f'(x)>0, the function ff is increasing; where f′(x)<0f'(x)<0, ff is decreasing. This conclusion concerns the original function, not the graph of its derivative.

  1. Find values where f′(x)=0f'(x)=0, where f′(x)f'(x) does not exist, and where ff is outside its domain.
  2. Use those values to divide the domain into open intervals.
  3. Determine the sign of f′f' on each interval, using a test value or algebra.
  4. Report the intervals with positive derivative as increasing and those with negative derivative as decreasing.

For f(x)=x3−3xf(x)=x^3-3x, f′(x)=3x2−3=3(x−1)(x+1).f'(x)=3x^2-3=3(x-1)(x+1). The derivative is zero at x=−1x=-1 and x=1x=1, so these values split the real line into three intervals.

Interval Sign of f′(x)f'(x) Behavior of ff
(−∞,−1)(-\infty,-1) ++ increasing
(−1,1)(-1,1) −- decreasing
(1,∞)(1,\infty) ++ increasing

Write increasing and decreasing sets as open intervals separated by critical values or domain breaks. A single point where f′(x)=0f'(x)=0 does not itself form an interval, and the sign must be established on each side rather than inferred from the zero alone.