Consider all rectangles lying in the region (x$$,$$y)∈ℝ×ℝ:$$0$$≤x≤π$$2$$ 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

Question

Consider all rectangles lying in the region (x$$,$$y)∈ℝ×ℝ:$$0$$≤x≤π$$2$$ 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

Infinity Learn · Accepted Answer

Perimeter $$=2(2$$α$$+2cos$$⁡$$2$$α$$)$$     since if $$x=$$π$$4+$$α⇒$$y=2$$ $$sin2$$$$π$$$$4+$$α$$=2$$ $$cos$$ $$2$$α$$P=4($$α$$+$$$$cos$$⁡$$2$$α$$)dPd$$α$$=4(1$$−$$2sin$$⁡$$2$$α$$)=0sin$$⁡$$2$$α$$=122$$α$$=$$π$$6$$,$$5$$$$π$$$$6d2Pd$$α$$2=$$−$$16cos$$⁡$$2$$α for maximum α$$=$$π$$12$$  if $$2$$α$$=$$π$$6$$⇒$$d2Pd$$α$$2=$$−$$16cos$$⁡π$$6=-83$$<$$0$$ Area $$=(2$$α$$)(2cos$$⁡$$2$$α$$)=$$π$$6$$×$$2$$×$$32=$$π$$23$$

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

$Consider all rectangles lying in the region$ $\{(x, y) \in ℝ \times ℝ : 0 \leq x \leq \frac{π}{2} and 0 \leq y \leq 2 \sin (2 x)\}$
$and having one side on the x-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is$

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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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Ready to Test Your Skills?

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$Consider all rectangles lying in the region$ $\{(x, y) \in ℝ \times ℝ : 0 \leq x \leq \frac{π}{2} and 0 \leq y \leq 2 \sin (2 x)\}$
$and having one side on the x-axis. The area of the rectangle which has the maximum perimeter among all such rectangles, is$