Lecture Note Differentiation
(^) Concave down
Concave up
f′′()x
xf )(
(^) |
(^) (-)^ (+) x
2
1
Test x= -1 Test x=1
f′′(1) 6(1) 3 9 0−=−−=−< f′′(1)=6 ( 1) 3 3+−=>^0
f is concave upward on
1
,
2
⎛⎞+∞
⎜
⎝⎠
⎟and downward on
1
,
2
⎛⎞−∞
⎜⎟
⎝⎠
b.
11
24
f
⎛⎞
⎜⎟=
⎝⎠
9
changes concavity at
1
2
x= , therefore the point
119
,
24
⎛
⎜ is a
point of inflection.
⎞
⎟
⎝⎠
Inflection point
119
,
24
⎛⎞
⎜⎟
⎝⎠
(0, 5)
9
1,
2
⎛⎞
⎜⎟
⎝⎠
Second-Derivative Test
Suppose thatfa′()= 0.
Iffa′′( )> 0 , then f has a relative minimum atx=a.
Iffa′′( )< 0 , then f has a relative maximum atx=a.
However, iffa′′( )= 0 , the test is inconclusive and f may have a relative
maximum, a relative minimum, or no relative extremum all at x=a.
a
x
y ()
()
0
0
fa
fa
′ =
′′ <
y ( )
()
0
0
fa
fa
′ =
a
′′ >
Relative maximum Relative minimum
x