Applied Mathematics for Business and Economics

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 .]

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| | | | | | | | | | | | | |

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