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Consider all rectangles lying in the region (x,y)×:0xπ2 and 0y2sin(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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By Expert Faculty of Sri Chaitanya
a
3π2
b
π
c
π23
d
π32
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detailed solution

Correct option is C

Perimeter =2(2α+2cos⁡2α)     since if x=π4+α⇒y=2 sin2π4+α=2 cos 2αP=4(α+cos⁡2α)dPdα=4(1−2sin⁡2α)=0sin⁡2α=122α=π6,5π6d2Pdα2=−16cos⁡2α for maximum α=π12  if 2α=π6⇒d2Pdα2=−16cos⁡π6=-83<0 Area =(2α)(2cos⁡2α)=π6×2×32=π23


ctaimg

Similar Questions

If f(x)=Min 2x2,x3:      2x<0 x|x|:       0x<1 3x:       1x2 now match the following 

Column IColumn II
A. The number of critical points of f(x) in [-2, 2] isP. 1
B. The length of the longest interval in which f(x) satisfies Rolle's theorem isQ. 2
C.The number of values of ‘c’ of Rolle's Mean value theoremR. 3
D. The maximum value of f(x) in [-2, 2] is S. 0

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