Analysis and Approaches for IBDP Maths Ebook 1 | Page 174

Your Practice Set – Analysis and Approaches for IBDP Mathematics
(c) f��
( x)
2
( x �1)(2) �(2 x)(2 x)
��
2 2
( x �1)
A2
2 2
2x
�2 �4x
��
2 2
( x �1)
(M1) for expansion
2
�2x
�2
��
2 2
( x �1)
2
2x
� 2
�
2 2
( x �1)
A1 N1
[4]
(d) f��( x) � 0
(M1) for setting equation
2
2x
� 2
( x �1)
2 2
� 0
2
2x �2 � 0
2( x�1)( x�1) � 0
x �1� 0 or x �1�
0
x �� 1 (Rejected) or x � 1
f (1)
2
� �ln(1 � 1)
(M1) for substitution
�� ln 2
A1
x x � 1 x � 1 x � 1
f�� ( x)
� 0 �
f�� ( x)
changes its sign at x � 1. (M1) for considering f��
( x)
Thus, the coordinates of the point of inflexion on
the graph of f are (1, � ln 2) for x � 0 . AG N0
[5]
(e) For concave downward on the left of (1, � ln 2) and
concave upward on the right of (1, � ln 2) A1
For passing through (0, 0)
A1
For decreasing behaviour A1 N3
[3]
A1
166
SE Production Limited