Lecture Note Differentiation
Example 2
Locate the local extrema for the function ()^34.
8
3
f xx= −x^
Solution
We Find f′()x and all values x=awhere fa′( )=0.
()
()
34
23
8
3
84
f xxx
f xxx
=−
′ =−
()
23
2
84 0
42 0
0or 2
xx
xx
xx
−=
−=
= =
Now, to apply the Second-Derivative Test, we find ff′′′′(0and)( (^2) ).
f′′(xx)=− 16 12 x^2
f′′(0 160120 0)=×−×=^2 the test fails.
f′′(2162122 16)=×−×=−^2
()
, indicating that f has a relative maximum at
x= 2. This value is^34
81
222
3
f
6
3
= ×−=
[The First-Derivative Test will show that f increases to the left of x = 0 and
to the right of x= 0. So x = 0 does not give a local minimum or a local maximum. The
test for concavity will show that a point of inflection occurs at x = 0 .]
| | | | | | | | | | | | | |
y
x
o,o
Point of
inflection
Local
maximum
16
2,
3
⎛⎞
⎜⎟
⎝⎠
43
3
8
)( −= xxxf
The behaviou of a graph when the first derivative is zero.
a
x
y
fa′()= 0
y
fa′′()= 0
a
x
( )
()
0
0
fa′ =
fa′′ =
Inflection Point Inflection Point