Increasing & Decreasing
Functions and 1st Derivative Test
Lesson 4.3
Increasing/Decreasing Functions
• Consider the following function
f(x)
a
• For all x < a we note that x1<x2 guarantees
that f(x1) < f(x2)
The function is said to be
strictly increasing
Increasing/Decreasing Functions
• Similarly -- For all x > a we note that x1<x2
guarantees that f(x1) > f(x2)
f(x)
The function is
said to be strictly
decreasing
a
• If a function is either strictly decreasing or
strictly increasing on an interval, it is said to
be monotonic
Test for Increasing and
Decreasing Functions
• If a function is differentiable and f ’(x) > 0 for
all x on an interval, then it is strictly
increasing
• If a function is differentiable and f ’(x) < 0 for
all x on an interval, then it is strictly
decreasing
• Consider how to find the intervals where the
derivative is either negative or positive
Test for Increasing and
Decreasing Functions
• Finding intervals where the derivative is
negative or positive
 Find f ’(x)
• f ‘(x) = 0
 Determine where
• f ‘(x) > 0
Critical
numbers
• f ‘(x) < 0
• f ‘(x) does not exist
1 3
f ( x)  x  9 x  2
3
• Try for
• Where is f(x) strictly increasing / decreasing
Test for Increasing and
Decreasing Functions
• Determine f ‘(x)
• Note graph
of f’(x)
• Where is
it pos, neg
f '( x)
f ( x)
f ‘(x) < 0 => f(x)f ‘(x)
decreasing
> 0 => f(x) increasing
f ‘(x) > 0 => f(x) increasing
• What does this tell us about f(x)
First Derivative Test
• Given that f ‘(x) = 0 at x = 3, x = -2,
and x = 5.25
• How could we find whether these points are
relative max or min?
• Check f ‘(x) close to (left and right) the point
in question
• Thus, relative minf ‘(x) < 0
f ‘(x) > 0

on left
on right
First Derivative Test
• Similarly, if f ‘(x) > 0 on left, f ‘(x) < 0 on right,

• We have a relative maximum
First Derivative Test
• What if they are positive on both sides of the
point in question?
• This is called an
inflection point

Examples
• Consider the following function
f ( x)  (2 x  1) ( x  9)
2
• Determine f ‘(x)
• Set f ‘(x) = 0, solve
• Find intervals
2
Assignment A
• Lesson 4.3A
• Page 226
• Exercises 1 – 57 EOO
Application Problems
• Consider the concentration
of a medication in the
bloodstream t hours after
ingesting
• Use different methods to determine when the
concentration is greatest
 Table
 Graph
 Calculus
3t
C (t ) 
27  t 3
t0
Application Problems
• A particle is moving along a line and its
position is given by s(t )  t 2  7t  10
• What is the velocity of the particle at t = 1.5?
• When is the particle moving in
positive/negative direction?
• When does the particle change direction?
Application Problems
• Consider bankruptcies (in 1000's) since 1988
1988
1989
1990
1991
1992
1993
1994
594.6
643.0
725.5
880.4
845.3
1042.1
835.2
• Use calculator regression for a 4th degree
polynomial
 Plot the data, plot the model
 Compare the maximum of the model, the
maximum of the data
Assignment B
• Lesson 4.3 B
• Page 227
• Exercises 95 – 101 all