Your Practice Set – Applications and Interpretation for IBDP Mathematics
43
Paper 1 – Second Derivative Test
Example
Let
3 2 f ( x) x 6x 15x � � � .
( a ) |
Find f�
( x)
.
|
( b ) |
Hence , solve the equation f�( x) �
0.
|
( c ) |
( i ) |
Write down f��
( x)
.
|
[ 2 ]
[ 2 ]
( ii ) Hence , determine the x -coordinate of the local maximum of f .
( iii ) Write down the y -coordinate of the local maximum of f . [ 4 ]
Solution
( a )
2
�( ) � 3 �6 ( 2 ) � 15 ( 1 ) ( A1 ) for correct approach f x x x
2 �( ) � 3 �12 � 15
A1 f x x x
( b ) f�( x) � 0
2
3x
�12x�15 � 0 3 ( x�5 )( x�1 ) � 0
( A1 ) for factorization x �� 5 or x � 1
( c ) ( i ) f ��( x) �6x
� 12
A1
A1
[ 2 ]
[ 2 ]
( ii ) f ��( �5 ) � 6 ( �5 ) � 12 f ��( �5 ) � �18 � 0
R1
Therefore , f attains its local maximum at x �� 5. Thus , the x -coordinate of the local maximum of f is � 5 .
A1
( iii ) 100 A1 [ 4 ]
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