What are Increasingand
Decreasing Function
• A function f is increasing on an interval if for
any two numbers x₁ and x2 in the interval, x1 <
x2 implies f(x₁) < f(x2).
• A function f is increasing on an interval if for
any two numbers x1 and x2 in the interval, x₁ >
x2 implies f(x₁) > f(x2).
• A function f is increasing on an interval if for
any two numbers X1 and X2 in the interval, x1 =
x2 implies f(x1) = f(x2).
3.
In the interval[a, b], the value of y decreases
as the value of x increases, hence, the
function y=f(x), is said to be decreasing in the
interval [a, b].
In the interval [b, c], the value of y increases
as the value of x increases, hence, the
function y = f(x), is said to be increasing in
the interval [b, c].
In the interval [c, d], the value of y decreases
as the value of x increases, hence, the
function y = f(x), is said to be decreasing in
the interval [c, d].
4.
Theorem 3.5 Testfor Increasing and
Decreasing Functions
Let f be a function that is continuous on the
closed interval [a, b]and differentiable on the
open interval (a, b).
1. If f'(x) > 0 for all x in (a, b), then f is increasing on [a, b].
2. If f'(x) < 0 for all x in (a, b), then f is decreasing on [a, b].
3. If f'(x) = 0 for all x in (a, b), then f is constant on [a, b].
5.
Guidelines for FindingIntervals on which a
Function is Increasing or Decreasing
Let f be continuous on the interval (a, b). To find the open
intervals on which fis increasing or decreasing, use the following
steps.
1. Locate the critical numbers of f in (a, b), and use these
numbers to determine test intervals.
2. Determine the sign of f'(x) at one test value in each of the
intervals.
3. Use Theorem 3.5 to determine whether f is increasing or
decreasing on each interval.
These guidelines are also valid if the interval (a, b) is replaced by an interval of the form (-00, b), (a, +00), or (-
00, +00).
6.
A different taxicompany displays their fare
prices in the table below:
Examples
1. What values of x
does the function y 9x
is increasing?
2. For intervalsof x is the
function y=3x-12x is increasing
or decreasing?
9.
Step 1: Findy'.
y = 3x²-12x
y'= 6x-12
Step 2: Set y'= 0.
6x-12-0
x=2 critical value is x = 2.
Step 3: Identify the intervals by the critical value.
Step 4: Check the sign of y' at an arbitrarily chosen
number x in each interval and apply Theorem 3.5.
Solution
10.
Find the openintervals on
which y = x³-12x-5 is
increasing or decreasing.
11.
Solution
Step 1: Findy
y=x-12x-5
y=3x²-12
Step 2: Set y'=0.
3x²-12=0
3(x²-4)=0
3(x-2)(x+2)=0
x-2=0 or x+2=0
x=2
Step 3: Identify the intervals by the
critical values.
Step 4: Check the sign of y' at an
arbitrarily chosen number x in each
interval and apply theorem 3.5.
![In the interval [a, b], the value of y decreases
as the value of x increases, hence, the
function y=f(x), is said to be decreasing in the
interval [a, b].
In the interval [b, c], the value of y increases
as the value of x increases, hence, the
function y = f(x), is said to be increasing in
the interval [b, c].
In the interval [c, d], the value of y decreases
as the value of x increases, hence, the
function y = f(x), is said to be decreasing in
the interval [c, d].](https://image.slidesharecdn.com/increasing-and-decreasing-function202502200809440000-250503230754-f3de2ac1/75/Increasing-and-Decreasing-Function_20250220_080944_0000-pptx-3-2048.jpg)
![Theorem 3.5 Test for Increasing and
Decreasing Functions
Let f be a function that is continuous on the
closed interval [a, b]and differentiable on the
open interval (a, b).
1. If f'(x) > 0 for all x in (a, b), then f is increasing on [a, b].
2. If f'(x) < 0 for all x in (a, b), then f is decreasing on [a, b].
3. If f'(x) = 0 for all x in (a, b), then f is constant on [a, b].](https://image.slidesharecdn.com/increasing-and-decreasing-function202502200809440000-250503230754-f3de2ac1/75/Increasing-and-Decreasing-Function_20250220_080944_0000-pptx-4-2048.jpg)
