Mathematics-I
Instructor: Sehrish Ijaz
Functions
Increasing and Decreasing Functions; Relative Extrema
Function 𝑓(𝑥) is increasing when the graph of 𝑓 is
rising.
𝑓(𝑥) is decreasing when the graph of 𝑓 is falling.
% of
GDP
1990 1991 1992 1993 1994 1995 years
Increasing Function
Let 𝑓(𝑥) be a function 𝒇(𝒙)
defined on the interval
𝒂 < 𝒙 < 𝒃 and 𝒙𝟏 + 𝒙𝟐
be the points in the interval
such that 𝑓 𝑥2
𝒙𝟐 > 𝒙𝟏
𝑓(𝑥) is increasing on the 𝑓 𝑥1
interval if
𝒇 𝒙𝟐 > 𝒇 𝒙𝟏 ; 𝒙𝟐 𝒙𝟏 .
a 𝑥1 𝑥2 b
Decreasing Function
If 𝑓(𝑥) be a function
defined on the interval
𝒂 < 𝒙 < 𝒃 and 𝒙𝟏 + 𝒙𝟐 𝒇(𝒙)
be the two numbers in the
interval such that
𝒙𝟐 > 𝒙𝟏 𝑓 𝑥1
𝑓(𝑥) is decreasing on the 𝑓 𝑥2
interval if
𝒇 𝒙𝟐 < 𝒇 𝒙𝟏
whenever 𝒙𝟐 > 𝒙𝟏 a 𝑥1 𝑥2 b
Increasing or Decreasing Derivatives
a b a b
Tangent or slope for 𝑓′(𝑥) is positive Tangent or slope for 𝑓′(𝑥) is negative
i.e. 𝒇′ 𝒙 > 𝟎 i.e. 𝒇′ 𝒙 < 𝟎
𝑓′(𝑥) is increasing on the interval 𝑓′(𝑥) is decreasing on the interval
where 𝒇′ 𝒙 > 𝟎 where 𝒇′ 𝒙 < 𝟎
Procedure for using the derivative to determine interval of increase and decrease
for a function 𝒇
Step 1
Find all the values of 𝑥 for which 𝒇′ 𝒙 = 𝟎 𝒐𝒓 𝒇′(𝒙) is not
continuous, and mark these numbers on a number line. This
divides the time into a number of open intervals.
Step 2
Choose a test number 𝑐 from each interval 𝒂 < 𝒙 < 𝒃
determined in step 1 and evaluate 𝑓′(𝑐). Then,
If 𝒇′ 𝒄 > 𝟎, the function 𝑓(𝑥) is increasing on 𝒂 < 𝒙 < 𝒃
If 𝒇′ 𝒄 < 𝟎, the function 𝑓(𝑥) is decreasing on 𝒂 < 𝒙 < 𝒃
Procedure for using the derivative to determine interval of increase
Limits
and decrease for a function 𝒇
E.g. 3.1.1
Find the intervals of increase or decrease for this function:
𝒇 𝒙 = 𝟐𝒙𝟑 + 𝟑𝒙𝟐 − 𝟏𝟐𝒙 − 𝟕
Notation
Up Arrow 𝑓(𝑥) increasing Down Arrow 𝑓(𝑥) decreasing
E.g. 3.1.2
Find the intervals of increase or decrease for this function:
𝒙𝟐
𝒇 𝒙 =
𝒙−𝟐
Relative Extrema
Peaks E H
A C
F
D
B
Valleys G
Relative Maxima: 𝑓(𝑥) is said to have relative maxima at 𝑥 = 𝑐 if 𝑓(𝑐) ≥ 𝑓(𝑥)
for all values of 𝑥 in an interval 𝑎 < 𝑥 < 𝑏
Relative Minima: 𝑓(𝑥) is said to have relative minimum at 𝑥 = 𝑐 if 𝑓(𝑐) ≤ 𝑓 𝑥
for all values of 𝑥 in an interval 𝑎 < 𝑥 < 𝑏
Critical Numbers and Critical Points
A number ‘c’ is called a critical number if either 𝑓 ′ 𝑐 = 0 𝑜𝑟 𝑓 ′ (𝑐)
does not exist. Point on the curve with coordinate (𝑐, 𝑓𝑐) is called a
critical point.
0(𝑐, 𝑓𝑐)
Relative extrema can only occur at a
critical point (where 𝑓 ′ 𝑥 = 0)
𝒇(𝒙) 𝑓′ 𝑐 = 0
a c b
Domain of 𝑥
Critical Numbers and Critical Points
𝒇′ < 𝟎
X
𝒇′ < 𝟎
𝒇′ < 𝟎 Z
𝒇′ > 𝟎
𝒇′ > 𝟎
Y
𝒇′ > 𝟎
A B C
(a) (b) (c)
Relative extreme occur at critical points, NOT ALL
CRITICAL POINTS corresponds to Relative Extrema.
The First Derivative Test for Relative Extrema
𝑓′ > 0
𝑓′ > 0 𝑓′ < 0
𝑓′ > 0 𝑓′ > 0
𝑓′ > 0
A Relative Maximum if 𝑓 ′ 𝑥 > 0 to
the left of 𝑐 and 𝑓 ′ 𝑥 < 0 to the right
of 𝑐.
𝑓′ > 0 c 𝑓′ < 0
The First Derivative Test for Relative Extrema
A Relative Minimum if 𝑓 ′ 𝑥 < 0 to
the left of 𝑐 and 𝑓 ′ 𝑥 > 0 to the right
of 𝑐.
𝑓′ > 0 c 𝑓′ < 0
Not a Relative Minimum if 𝑓 ′ 𝑥 has
the same sign o both sides of 𝑐.
𝑓′ > 0 c 𝑓′ < 0 𝑓′ > 0 c 𝑓′ < 0
Examples
E.g. 3.1.3
Find all the critical numbers of the function
𝒇 𝒙 = 𝟐𝒙𝟒 − 𝟒𝒙𝟐 + 𝟑
and classify each critical point at a Relative
Maximum, a Relative Minimum or neither.
Procedure for sketching the graph of a continuous function 𝒇(𝒙) using the
derivative 𝒇′(𝒙)
Step 1:
Determine the domain of 𝑓(𝑥). Setup a number line restricted to include only
those numbers in the domain of 𝑓 𝑥 .
Step 2:
Find 𝑓′(𝑥) and mark each critical number on the number line and analyze the
sign of the derivative.
Step 3:
For each critical number 𝑐, find 𝑓(𝑐) and plot the critical point 𝑃(𝑐, 𝑓(𝑐)) with a
cap ( ) or cap ( )
Step 4: Sketch the graph as smooth curves joining the critical points.
Examples
E.g. 3.1.4
Sketch the graph of the function:
𝒇 𝒙 = 𝒙𝟒 + 𝟖𝒙𝟑 + 𝟏𝟖𝒙𝟐 − 𝟖
