1. 5-3 Day 1 connecting f graphs with f' and f" graphs

2. Remember… your calculator is to CONFIRM and support – NOT to do the entire question!
Thm: First Derivative Test for Local Extrema
(1) Local Max  at critical point c
a) f '(c) = b) f '(c) is undefined c c
f ' > 0
f ' < 0
f ' > 0
f ' < 0
What is happening? rising falling
local max

3. (2) Local Min  at critical point c
a) f '(c) = b) f '(c) is undefined c c
f ' < 0
f ' > 0
f ' < 0
f ' > 0
What is happening? falling rising
local min
(3) Nothing  at critical point c
a) f '(c) = b) f '(c) is undefined c c
f ' > 0
f ' > 0
f ' < 0
f ' < 0

4. At a LEFT endpoint a
a) Local Max
b) Local Min a a
f ' < 0
f ' > 0
At a RIGHT endpoint b
a) Local Min
b) Local Max b b
f ' < 0
f ' > 0

5. Using the First Derivative Test 
Find deriv & critical values
Put critical values on a # line (deriv)
Find if each interval is (+) or (–)
Where it happens (+)(–) (–)(+) (+)(+) or (–)(–)
max min nothing
Note: To find the coordinate, plug x-values into original function

6. Ex 1) For each of the following functions, use the First derivative Test to find the local extreme values. Identify any absolute extrema. a) + – +
f ' –2 2
max
min
(–2, 11)
Min @ (2, –21)
Range (–, )
 No abs max/min

7. Ex 1) For each of the following functions, use the First derivative Test to find the local extreme values. Identify any absolute extrema. b) + – +
Min @ g' –3 1
max
min
Abs min at No abs max

8. Concavity
y' increases if y" > 0
y' decreases if y" < 0
Concavity Test: Let y = f (x),
a) Concave up where y" > 0
b) Concave down where y" < 0
Ex 2) Use the Concavity Test to determine the concavity of the given function in the given interval.
b) y = 3 + sin x on (0, 2)
y' = cos x
y" = – sin x = 0
x = 0, , 2 – +
y"  2
Concave down (0, )
Concave up (, 2)

9. Points of Inflection
The point where the graph has a tangent & concavity changes
Where y" = 0 or is undefined
How? Sign chart for y"
Ex 3) Find all points of inflection of the graph of + – +
y"
Pts of Infl:

10. Ex 4) The graph of the derivative of a function f on the interval [–4, 4] is shown. Answer the following questions about f, justifying each answer with information obtained from the graph of f '.
f '
On what intervals is f increasing?
(–4, 1)
On what intervals is f decreasing?
(1, 4)
Where are the local extrema?
x = 1 &
x = –4 & x = 4 + + –
f ' –4 –2 1 4

11. homework
Pg # 19, 29, 44
Pg # 1–6, 9, 12, 15, 21, 24