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Question

The minimum area bounded by the function y=f(x) and y=αx+9 (αϵR) where f satisfies the relation f(x+y)=f(x)+f(y)+yf(x) x,yϵR and f(0)=0 & f(0)=0 is 9A, value of A is ___

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Solution

f(x)=limh0f(x+h)f(x+0)h=limh0f(x)+f(h)+hf(x)f(x)f(0)0f(x)h

=limh0(f(h)f(0)h0)+f(x)

=limh0(f(h)1)+f(x)

f(x)=f(x)

f(x)f(x)dx=dx

2f(x)=x+c

f(x)=x24

when α=0 area is minimum

required minimum area = 2902ydy

4(y3232)90 = 72 sq.unit


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