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Paper 2 – Optimization Problems
Example
The following diagram shows a quarter of a circle with centre O and radius 2 . Let P be a point on the circumference and Q be a point on the x -axis such that PQ and OQ are
perpendicular , with segment .
ˆ POQ �
� � radians , where 0 ��
� . Let S be the area of the shaded 2
( a ) Find the area of the triangle OPQ in terms of � .
( b ) Express S in terms of � and � .
( c ) ( i ) Find d S
d� .
( ii ) Solve the equation d S
0 d� � .
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It is given that
2 d S d
8sin�cos�
2
� � .
( iii ) Hence , find the value of � when S is at a local minimum , justifying that it is a minimum .
( iv ) Using ( b ) and ( c )( iii ) to find the minimum value of S .
( d ) Write down the two values of � for which S has its greatest value .
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