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Question

Let A(k) be the area bounded by the curves y=x23 and y=kx+2. Then

A
the range of A(k) is [1053,)
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B
the range of A(k) is [2053,)
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C
If function kA(k) is defined for k[2,), then A(k) is many- one function
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D
the value of k for which area is minimum is 1
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Solution

The correct options are
B the range of A(k) is [2053,)
C If function kA(k) is defined for k[2,), then A(k) is many- one function

Line y=kx+2 passes through fixed point (0,2) for different value of k.

Also, the minimum A(k) occurs when k=0, as when line is rotated from this position about point (0,2), the increased part of area is more than the decreased part of area.

A(k)min=250[2(x23)]dx
=2[5xx33]50
=2053 sq. units

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