Analysis and Approaches for IBDP Maths Ebook 1 | Page 168

Your Practice Set – Analysis and Approaches for IBDP Mathematics
56
Paper 2 Section B – Gradients
of the graphs of functions
Example
3 2
Let f ( x) � ax �bx � 4bx � b, where a and b are real numbers. The graph of f passes
through the point (6,135) .
(a) Show that 216a�11b� 135 .
The graph of f has a local minimum at (6,135) .
[2]
(b) Find the other equation in a and b , giving your answer in a similar form to part
(a).
[5]
(c) Find the value of a and of b .
[3]
(d) Find the coordinates of the local maximum point of the graph of f .
[4]
Solution
(a) f (6) � 135
(M1) for valid approach
(b) f�
( x)
(c)
3 2
a(6) �b(6) � 4 b(6) �b
� 135
A1
216a �36b � 24b �b
� 135
216a�11b� 135
AG N0
2
� 3ax � 2bx � 4b
A2
f �(6) � 0
(M1) for valid approach
2
3 a(6) 2 b(6) 4b
0
� � � (A1) for substitution
108a�8b�
0
27a�2b� 0
A1 N3
[5]
�216a�11b�135
�
�27a�2b�0
(M1) for solving the system
a � 2 , b �� 27
A2 N3
[3]
[2]
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