Analysis and Approaches for IBDP Maths Ebook 2 | Page 151

2. The function f is defined by
1
x
2
e �
f ( x) �2
� , x� .
2
11
(a) Write down the equation of the horizontal asymptote of f( x ).
(b) Find f� ( x)
.
(c) Show that there is no local maximum of f( x ) for x� .
(d) Find the range of f( x ).
(e) Find f�� ( x)
.
(f) Show that there are two points of inflexion of f( x ) for x� .
(g) Let g( x) � e f ( x)
� k . It is given that the points of inflexion of gx ( ) lies on the
x -axis. Find the exact value of k .
x
3. The function f is defined by f( x)
�
2
x � 36
, x �� 6 , x � 6 .
(a) Write down the equations of the vertical asymptotes of f( x ).
(b) Find f� ( x)
.
(c) Show that there is no local extrema of f( x ).
(d) Find f�� ( x)
.
(e) Show that there is only one point of inflexion of f( x ).
(f) Let gx ( ) be another function. By expressing f( x ) in partial fractions, find the
expression of gx ( ) if f( x ) is the sum of gx ( ) and gx� ( 12) .
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