3.3
st
1
Derivative Test
Increasing or Decreasing?:
a) If f (x) > 0 in an interval,
then f is increasing in the
interval.
b) If f (x) < 0 in an interval,
then f is decreasing in the
interval.
1st Derivative Test
c is critical number of f:
a) If f changes from + to –
at c, then f(c) is a local
max.
b) If f changes from – to +
at c, then f(c) is a local
min.
Ex 1: Find where f (x) is
increasing or decreasing:
f x   3x  4 x  12 x  5
4
3
2
Ex 2: Find the local min &
local max values of the
function:
f x   x
2
3
6  x
1
3
HW – 3.3 pg. 186
#1 – 7 odds,
#17 – 45 EOO,
#55 – 63 odds
3.4 Concavity Test
Concave Up or Down?:
a) Concave up: holds water
• Inc @ an Increasing
rate
• Dec @ a Decreasing
rate
Concave Up or Down?:
b) Concave down: spills
water
• Inc @ a Decreasing
rate
• Dec @ an Increasing
rate
Concavity Test
a) f (x) > 0 in an interval,
then f is concave up in
the interval.
b) f (x) < 0 in an interval,
then f is concave down in
the interval.
Point of Inflection
 point where f changes
concavity.
 where f  changes from
increasing to decreasing or
vice versa.
 where f  changes sign.
Ex 1: Find where f (x) is
concave up or concave
down:
f x   3x  4 x  12 x  5
4
3
2
Ex 2: Find the points of
inflection of the function:
f x   x
2
3
6  x
1
3
HW – 3.4 pg. 195
# 1 – 5 odds,
#11 – 39 EOO,
#49 – 56 all
2nd Derivative Test
c is critical number of f:
a) If f (c) = 0 & f (c) > 0,
then f(c) is a local min.
b) If f (c) = 0 & f (c) < 0,
then f(c) is a local max.