Analysis and Approaches for IBDP Maths Ebook 2 | Page 169

2 1
4. The function gx ( ) is defined by g( x)
� e �x , 0 � x � . Let B� and C� be the points on
2
1
the x -axis such that the x -coordinates of B� and C� are h and
2 respectively,
1
0 �h
� . Let A , B and C be the points on the graph of gx ( ) such that AO , BB�
2
and CC� are vertical lines.
11
(a) Find g�� ( x)
.
(b) Show that the graph of gx ( ) is concave downward for
Let T be the sum of the areas of trapeziums OABB� and B�BCC
�.
1
0 � x � .
2
[3]
[2]
(c) (i) Express T in terms of h .
(ii)
Express d T
dh in terms of h .
(iii)
Hence, determine the maximum value of T , justifying that this value
is a maximum.
(d)
(iv) Write down the value of h when T attains its maximum.
[12]
2�
1 �
Show that T � 1
4
� � �
� e � . [2]
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