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1 f ’’(x) > 0 for all x in I
f(x) concave Up
Concavity Test
Sec 4.3: Concavity and the Second Derivative Test
the curve lies above the tangents the curve lies below the tangents
2 f ’’(x) < 0 for all x in I
f(x) concave Down
Concavity Test
1 f ’’(x) > 0 for all x in I
2
f(x) concave Up
Concavity
f ’’(x) < 0 for all x in I f(x) concave Down
Sec 4.3: Concavity and the Second Derivative Test
Example:Study the concavity of the function
34 4)( xxxf
Sec 4.3: Concavity and the Second Derivative Test
F092
Sec 4.3: Concavity and the Second Derivative Test
1
2
Inflection point:
v.tangenthas or exists )(' fcf
c''f +_
point inflection
(c,f(c))
_+
Sec 4.3: Concavity and the Second Derivative Test
Sec 4.3: Concavity and the Second Derivative Test
Find all inflection points
Example:Find all inflection points of
34 4)( xxxf
Sec 4.3: Concavity and the Second Derivative Test
8
F091
Sec 4.3: Concavity and the Second Derivative Test
Sec 4.3: Concavity and the Second Derivative Test
1
2
second Derivative Test:
number critical is c
)('' cfc
f
at minimum
local has
1
2
second Derivative Test:
number critical is c
)('' cfc
f
at max
local has
Sec 4.3: Concavity and the Second Derivative Test
cf containing Ion cont ''
cf containing Ion cont ''
1
2
second Derivative Test:
number critical is c
0)('' cf
cf containing Ion cont ''
the test fails. The function ƒ may have a local maximum, a local minimum, or neither.
1
2
number critical is c
)('' cf
cf containing Ion cont ''
1
2
number critical is c
)('' cf
cf containing Ion cont ''
1
2
number critical is c
0)('' cf
cf containing Ion cont ''
F121
Sec 4.3: Concavity and the Second Derivative Test
F081
Sec 4.3: Concavity and the Second Derivative Test
Sec 4.3: Concavity and the Second Derivative Test
Sec 4.3: Concavity and the Second Derivative Test
