Increasing & Decreasing Functions; First Derivative Test

Definitions

Increasing Intervals:     x1 < x2 implies f(x1) < f(x2)

Decreasing Intervals:    x1 < x2 implies f(x1) > f(x2)

This function is increasing over -oo < x < -2 and
                                                               3 < x < oo and is decreasing over -2 < x < 3

Test for Increasing or Decreasing Intervals

Proof:

Assume f '(x) > 0 for all x in (a,b).

     I.  Let f '(x) > 0 for all x in (a,b).

          Let x1 < x2
            The Mean Value Theorem says
             f '(c) = f(x2) - f(x1)
                             x2 - x1

            Since f '(x) > 0 and x2 - x1 > 0, then f(x2) - f(x1) > 0
          Therefore f(x2) > f(x1) and f(x) is increasing

    II.  Let f '(x) < 0 for all x in (a,b).

          The proof is similar to the proof in I.

First Derivative Test

  1. If f '(x) changes from negative to positive at c, then f(c) is a relative minimum of f(x).
  2. If f '(x) changes from positive to negative at c, then f(c) is a relative maximum of f(x).
  3. If f '(x) does not change signs at c, then f(c) is neither a relative minimum nor a relative maximum.
Proof:

     I.  Assume f '(x) changes from negative to positive at c.
         Then f '(x) < 0 for all x in (a,c) and f '(x) > 0 for all x in (c,b).
         Thus f(x) decreases on (a,c) and increases on (c,b).
         Therefore, f(c) is a minimum.

   II.  Assume f '(x) changes from positive to negative at c.
         The proof is similar to the proof in I.

Definition

A function is strictly monotonic on an interval if it is either increasing on the entire interval or decreasing on the entire interval.


Problems