Analysis and Approaches for IBDP Maths Ebook 2 | Page 150

Your Practice Set – Analysis and Approaches for IBDP Mathematics
(f) f��( x) � 0
�
x
3
2 2
(1 � x )
�0
x � 0
x x � 0 x � 0 x � 0
f�� ( x)
� 0 �
f�� ( x)
changes its sign at x � 0 . M1
(M1) for setting equation
f (0) �arcsin 0 � 4(0)
(A1) for substitution
f (0) � 0
Thus, the coordinates of the point of inflexion of
f( x ) are (0, 0) .
(g) f�( x) � 0 for �1�
x � 1
A1
Thus, f( x ) is strictly decreasing for �1� x � 1. R1
� f (1) � f ( x) � f ( � 1)
M1
� f ( �1) � f ( x) � f ( � 1)
A1
f ( x) � f ( � 1)
π
f( x) �4 � AG
2
[4]
[3]
Exercise 46
1. The function f is defined by
2
f ( x) � arctan x � ln(1 � x ) , x� .
(a) Find the value of a such that f ( a) � arctan a .
(b) Find f� ( x)
.
(c) Show that there is no local maximum of f( x ) for x� .
(d) Find the coordinates of the local minimum of f( x ).
(e) Find f�� ( x)
.
(f) Show that there are two points of inflexion of f( x ) for x� .
[4]
(g) It is given that � 1.17 and 0 are the solutions of the equation f( x) � 0 . Solve the
inequality f( �x) � 0.
[2]
[2]
[3]
[2]
[2]
[3]
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