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If f be decreasing continuous function s...

If `f` be decreasing continuous function satisfying `f(x + y) = f(x) + f(y)-f(x) f(y) Y x, y epsilon R ,f'(0)=1` then `int_0^1 f(x)dx` is

A

1

B

`1-e`

C

`2-e`

D

none of these

Text Solution

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The correct Answer is:
C
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Knowledge Check

  • Let f be a continuous function satisfying f(x + y) = f (x) f( y) (x, y in R) with f(1) = e then the value of int(xf(x))/(sqrt(1+f(x)))dx is

    A
    `2xsqrt(1+e^(x))-4sqrt(1+e^(x))-2log|((sqrt(1+e^(x))+1)/(sqrt(1+e^(x))-1)) |+C`
    B
    `2sqrt(1+e^(x))-4sqrt(1+f(x))-2log|sqrt(1+f(x))-1|+C`
    C
    `2sqrt(1+f(x))-4sqrt(1+f(x))-2log|(sqrt(1+f(x))-1)/(sqrt1+f(x))|+C`
    D
    none of these

  • Let f be a continuous function satisfying f (x+y) = f (x) + f (y), for each x, y in R and f(1) = 2 then int(f(x)tan^(-1)x)/((1+x^(2))^(2))dx is equal to

    A
    cannot be determined explicity
    B
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    C
    `C-(1)/((1+x^(2)))tan^(-1)x+(1)/(2)tan^(-1)x+(x)/(2(1x^(2)))`
    D
    `C-(1)/((1+x^(2)))tan^(-1)x+(1)/(2)tan^(-1)x+(x)/(2(1+x^(2)))`

  • IF f(x+f(y))=f(x)+y AA x, y in R and f(0)=1 , then int_(0)^(10)f(10-x)dx is equal to

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    C
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    D
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