1. W-up 3-27 When is the function increasing?
For the interval(s) that it is increasing, what do all the derivatives have in common?
When is the function decreasing?
For the interval(s) that it is decreasing, what do all the derivatives have in common?
State local max and min
What do you know about the derivative at the local max and mins?

2. 14.2 Increasing/Decreasing functions; The First Derivative Test
Determine where a function is increasing and where it is decreasing
Use first derivative test
Graph functions

3. Increasing/ Decreasing
Find the derivative
Set up a table that solves f ‘(x) >0 and f ‘(x) < 0
It’s increasing on the interval (a,b) if f ‘(x) >0
It’s decreasing on the interval (a,b) if f ‘(x) < 0

4. Ex: Determine when f(x) = x3 – 6x2 +9x – 2 is increasing or decreasing
Find derivative: f’(x) = 3x2 – 12x + 9
Set derivative = 0 and solve
3x2 – 12x + 9 = 0
3(x2 – 4x + 3) = 0 {factored out a GCF}
3(x – 1)(x – 3) = 0
x = 1 and x = 3
Set up table to determine when positive/negative 1 3
Interval
f’(x)
Inc/dec
(-∞,1)
(1,3)
(3,∞)
Pick a point less than 1
f’(0) = 9 positive
Pick a point btw
1 & 3
f’(2) = negative
Pick a point greater than 3
f’(4) = positive
Decreasing
Increasing
Increasing

5. 1st derivative test: - take the first derivative and…
A - MAX
If f is increasing to the left of a point A on the graph f and is decreasing to the right of A, there is a local MAXIMUM
If f is decreasing to the left of a point B on the graph f and is increasing to the right of A, there is a local MINIMUM
increasing
decreasing
increasing
decreasing
B - MIN

6. steps to graph a function
Find domain
Located intercepts (if reasonable to find)
Locate all points where tangent line is vertical or horizontal f’(x) = 0
Determine where graph is increasing and decreasing
Find local max/min using 1st derivative test
Determine end behavior, locate any asymptotes

7. Ex: follow 6 steps graph f(x) = x3 – 12x
Domain – all real numbers
x- intercepts - factor and solve
x(x2 – 12)=0
x = 0 x2 – 12 = 0
x = ± 𝟏𝟐 =±𝟐 𝟑
3-5)Increasing/decreasing/1st derivative test/ horizontal tangent line, no vertical tangent lines
f’(x) = 3x2 – 12
0 = 3x2 – 12
3(x2 – 4) =0
x =±𝟐
f(2) = 23 – 12(2) = f(-2) = -23 – 12(-2) = 16
horizontal tangent lines at (2,-16) and (-2,16)

8. f(x) = x3 – 12x
f’(x) = 3x2 – 12
MAX
Plot points of the horizontal tangent line(s)
Make a table to determine increasing/decreasing
Plot x-intercepts
End behavior – acts like x3
MIN
Pick a point less than -2 say -3
f’(-3) = positive
f(x) is INCREASING
(-∞,−2)
Pick a point between - 2& 2 say 0
f’(0) = negative
f(x) is DECREASING
(-2,2)
Pick a point greater than 2 say 3
f’(3) = positive
f(x) is INCREASING
(2,∞)

9. 14.2 hw # 1-8 all #9-21 odds (lets do #17 in class)
#43-46 (read the notes on homework paper)