Download Positive

3 2
yx  x
2
3
Once I find a critical
number, how do I know if a
maximum or a minimum
occurs at the point?
We will examine intervals of “Increasing”
and “Decreasing” for a function and
develop a “First Derivative Test”
3 2
yx  x
2
3
Here are the intervals of
increasing and decreasing
What is the “slope” when a
function is increasing? Positive
What is the “slope” when a function is
decreasing? Negative
3 2
yx  x
2
3
I
II
III
To find the intervals of Inc/Dec
1)Find the critical numbers of f, this
will find your test intervals
2)Find the sign of f’(x) for one value
in each interval
3)If f’(x) > 0, it is Inc. in the interval
If f’(x) < 0, it is Dec. in the interval
Find the intervals of Inc/Dec
1) Find critical numbers, f’(x) = 0
y '  3x  3x  0
2
3 x( x  1)  0
x  0,1
3 2
yx  x
2
3
This gives the test intervals:
(0, 1)
(1, ∞)
(-∞, 0)
0
Now test an x-value from each interval
using the derivative
(0, 1)
(-∞, 0)
(1, ∞)
0
I) In (-∞, 0) Chose x = -1
Evaluate in the first derivative
f ' ( x)  3 x  3 x
2
f ' (1)  3(1)  3(1)  6
2
This is positive so the function is
increasing in this interval
(-∞, 0)
(0, 1)
(1, ∞)
0
II) In (0, 1) Chose x = 1/2
Evaluate in the first derivative,
here I will use the factored form
f ' ( x)  3x( x  1)
f ' ( 1 )  3( 1 )( 1  1)  ()( )  ()
2
2 2
It is negative at this point, so the function is
decreasing in the interval
(-∞, 0)
(0, 1)
(1, ∞)
0
III) In (1, ∞) Chose x = 2
Evaluate in the first derivative, again I
will use the factored form
f ' ( x)  3x( x  1)
f ' (2)  3(2)( 2  1)  ()( )  ()
It is positive at this point, so the function is
increasing in the interval
(-∞, 0)
(0, 1)
(1, ∞)
0
Now that we know the intervals of Inc/Dec, we can
apply the First Derivative Test by examining the
behavior of the graph surrounding a critical
number
Around x = 0, the function “peaks”, so
it is a Relative Maximum
Around x = 1, the function has a
“trough”, so it is a Relative Minimum