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Let f be a derivable function on [0, 3] ...

Let `f` be a derivable function on `[0, 3]` such that `f'(x) leq 1 AA x in [0,3]` where `f(0)=0 & f(3)=3` then `f(1)=`

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Knowledge Check

  • Let f(x) be a derivable function, f'(x) gt f(x) and f(0)=0 . Then

    A
    `f(x) gt 0" for all " x gt 0`
    B
    `f(x) lt 0" for all " x gt 0`
    C
    no sign of f(x) can be ascertained
    D
    f(x) is a constant function

  • Let f(x) be a fourth differentiable function such that f ( 2 x^(2) - 1) =2 x f (x) AA x in R then f^(iv) (0) is equal to (where f^(iv) (0) represents fourth derivative of f(x) at x = 0 )

    A
    0
    B
    1
    C
    `-1`
    D
    Data insufficient

  • Let f (x) be function defined on [0,1] such that f (1)=0 and for any a in (0,1], int _(0)^(a) f (x) dx - int _(a)^(1) f (x) dx =2 f (a) +3a +b where b is constant. int _(0)^(1) f (x) dx =

    A
    `1/e`
    B
    `(1)/(2e)`
    C
    `(3)/(2e)`
    D
    `(2)/(e)`

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    Let f(x) be a differentiable function satisfying f(y)f((x)/(y))=f(x)AA,x,y in R,y!=0 and f(1)!=0,f'(1)=3 then

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    Let f (x) be function defined on [0,1] such that f (1)=0 and for any a in (0,1], int _(0)^(a) f (x) dx - int _(a)^(1) f (x) dx =2 f (a) +3a +b where b is constant. The length of the subtangent of the curve y= f (x ) at x=1//2 is:

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