Calculus I
Chapter 4(2)
Increrasing
Decreasing
First Derivative Test
Increasing & Decreasing Functions
A function is Increasing between two points if
the y-value of the second point is greater than the
y-value of the first point.
A function is Decreasing between two points if
the y-value of the second point is lower than the
y-value of the first point.
Give the intervals where the function is
increasing, decreasing, and constant
a
b
c d
e
f g
h
i
Increasing intervals: (a, b)  (c, e)  ( g , h)  (i, j)
Decreasing Intervals: (b, c)  ( f , g )  (h, i)
Constant Intervals: (e, f )
j
Derivatives and increasing &
Decreasing Functions
A function is Increasing at a point if the derivative
is positive at the point (tangent line is going up)
A function is Decreasing at a point if the derivative
is Negative at the point (tangent line is going down)
Steps for finding where a function is
increasing & decreasing
Find the Critical Values (when f’(x) = 0)
Graph the Critical values on a number line
Check the intervals in f’(x) to see if
Positive – Increasing
Negative - Decreasing
Find the increasing intervals and the decreasing
2
intervals for: f ( x)  8x  2 x
Find Critical Values
(derivative = 0)
f '( x)  8  4 x
2
8  4 x  0 so x  2
Put Critical Value(s) on a number Line:
F(x) is Increasing
Put a # from the left side
(1) into the derivative
f (1)  8  4(1)  Positive
Positive means increasing
F(x) is Decreasing
2
Put a # from the right side
(3) into the derivative
f (3)  8  4(3)  Negative
Negative means Decreasing
(-,2) is Increasing, (2, ) is Decreasing
First Derivative Test
If f’(x) changes from negative to positive at
some point c, then f(x) is a relative minimum.
Positive
Derivative
Negative
Derivative
Relative Minimum
If f’(x) changes from positive to negative at
some point c, then f(x) is a relative maximum.
Relative Maximum
Positive
Derivative
Negative
Derivative
Steps for finding relative max & min
Find the Critical Values (when f’(x) = 0)
Graph the Critical values on a number line
Check the intervals in f’(x) to see if
Positive – Increasing
Negative - Decreasing
Increasing to Decreasing - Max
Decreasing to Increasing - Min
Find the local extrema, the increasing intervals and
the decreasing intervals for: f ( x)  x3  6x2  15
3x  12 x  0
2
Find Critical Values
3x( x  4)  0 so x  0 & 4
(derivative = 0)
Put Critical Value(s) on a number Line:
F(x) is Increasing
Pick -1
0
f (1)  ( Neg )( Neg )
 Positive
F(x) is Decreasing
Pick 1
F(x) is Increasing
4
Pick 5
f (1)  ( Pos)( Neg ) f (5)  ( Pos)( Pos)
 Positive
 Negative
(-,0) U (4,) is Increasing, (0, 4) is Decreasing
(0, 15) Max, (4, -17) Min
Find the local extrema, the increasing intervals and
the decreasing intervals for: f ( x)  x  3
Find Critical Values
x2
x 2 (1)  ( x  3)(2 x)
0
2 2
(x )
( x  6)
 0 so x  6 &  0
3
x
(derivative = 0)
Put Critical Values on a number Line:
F(x) is Decreasing
Pick -7
f (7) 
F(x) is Increasing
-6
Neg ( Neg )
Neg
 Negative
Pick -1
f ( 1) 
Neg ( Pos )
Neg
 Positive
F(x) is Decreasing
0
Pick 1
f (7) 
Neg ( Pos )
Pos
 Negative
(-,-6) U (0,) is Decreasing, (-6, 0) is Increasing
x = 0 Vertical Asymptote, (6, -1/12) Min
Concavity
A graph is Concave Up on an interval if f’ is
increasing on the interval.
A graph is Concave Down on an interval if f’ is
decreasing on the interval.
f(x)
Draw the derivative graph and indicate where the
graph is concave up and concave down and
where it is increasing and decreasing.
f(x)
a
b
c
d
e
f
g
f ' ( x)
Concave Up (a, c)
Concave Down (c, e) U (e,f)
Increasing (b, d) U (e ,f)
Decreasing (a, b) U (d ,e) U (f ,g)
Second Derivative and Concavity
A graph is Concave Up on an interval if f ( x)
is positive on the interval.
A graph is Concave Down on an interval if f ( x)
is negative on the interval.
f(x)
f ( x)
Inflection Point
An Inflection Point is a point where the
concavity changes
This is where the second derivative = 0 or is undefined
Steps for finding where a function is
Concave Up or Down
Find the second derivative and set it = 0
Graph the resulting values on a number line
Check the intervals in f ( x) to see if
Positive – Concave up
Negative – Concave Down
Find the inflection points and where concave
up & down:
f ( x)  e x / 2
2
Find Second
Derivative Values
f ( x)   xe
 x2 / 2
f ( x)  e
 x2 / 2
Inflection
( x2 1) so x  1
Put Inflection points on a number Line:
Concave Up
Pick -2
f (2)  ( Pos)( Pos)
 Positive
Concave Down
-1
Pick 0
Concave Up
1
Pick 2
f (0)  ( Pos)( Neg ) f (2)  ( Pos)( Pos)
 Positive
 Negative
(-,-1) U (1, ) is Concave Up, (-1,1) is Down
Second Derivative Test
If f (c)  0 for some value of c, and f (c)  0
then c is a minimum point on the graph of f(x)
Concave Up
Relative Minimum
If f (c)  0 for some value of c, and f (c)  0
then c is a Maximum point on the graph of f(x)
Relative Maximum
Concave
Down
Find the Extrema and inflection points and give the
5
3
f ( x)  3x  5x
concavity intervals for:
Find both
Derivatives
and set =0
f ( x)  15x4  15x2
 15x2 ( x2 1) so x  0, 1
3

f ( x)  60 x  30 x
f ( x)  30x(2x2 1)
x0 & 
1
2
Put all Points on a number Line and finish: