Q.
Consider all rectangles lying in the region (x,y)∈ℝ×ℝ:0≤x≤π2 and 0≤y≤2sin(2x)and having one side on the x-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is
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a
3π2
b
π
c
π23
d
π32
answer is C.
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Detailed Solution
Perimeter =2(2α+2cos2α) since if x=π4+α⇒y=2 sin2π4+α=2 cos 2αP=4(α+cos2α)dPdα=4(1−2sin2α)=0sin2α=122α=π6,5π6d2Pdα2=−16cos2α for maximum α=π12 if 2α=π6⇒d2Pdα2=−16cosπ6=-83<0 Area =(2α)(2cos2α)=π6×2×32=π23
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