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Question

Assertion :

Consider the two curves y=xλx2 and y=x2λ,(λ>0)
The area bounded between the curves is maximum when λ=1 Reason: The area bounded between the curves is λ2(1+λ2)2 square units

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Both Assertion and Reason are incorrect
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Solution

The correct option is C Assertion is correct but Reason is incorrect
We obtain point of intersection of two curves by equating them
xλx2=x2λ
λxλ2x2=x2
x2(1+λ2)λx=0
x[x(1+λ2)λ]=0
x=0 or x=λ1+λ2
Let λ1+λ2=k
Hence, the required area will be
k0x2λx+(λ)x2 dx
=1λk0x2(1+λ2)λx dx
=1λ[x33(1+λ2)λx22]k0
=1λ[k33(1+λ2)λk22]
=1λ[λ33(1+λ2)2λ32(1+λ2)2]
=λ26(1+λ2)2
Hence, area =16(λ1+λ2)2
Hence, area is maximum at λ=1.

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