Apps. and Interpretation for IBDP Maths Ebook 1 | Page 207

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Paper 2 – Analysis of Graphs of Functions
Example
Let
3 2
f ( x) � 2x � 27x �108x
� 27 . The graph of f has a local minimum at (6, r ) .
(a) Find the value of r .
(b) Find f� ( x)
.
(c) Find the coordinates of the local maximum point of the graph of f .
(d) Write down the interval of x when f( x ) is decreasing.
(e) (i) Write down the value of f (4) .
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(ii) Find the value of f � (4) .
(iii) Hence, find the equation of tangent at x � 4 , giving the answer in general
form.
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Solution
(a)
(b)
3 2
r � 2(6) � 27(6) �108(6) � 27
(A1) for substitution
r � 135
A1 N2
2
�( ) �2(3 ) �27(2 ) �108(1) � 0
(A1) for correct derivatives
f x x x
2
�( ) � 6 �54 � 108
A1 N2
f x x x
(c) f�( x) � 0
2
6x
54x
108 0
� � � (M1) for setting equation
6( x�3)( x�6) � 0
x � 3 or x � 6 (Rejected) (A1) for correct value
3 2
f (3) � 2(3) � 27(3) �108(3) � 27
f (3) � 162
Thus, the required coordinates are (3,162). A1 N3
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