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Let the set of all function f: [0,1]toR,...

Let the set of all function `f: [0,1]toR,` which are continuous on `[0,1]` and differentiable on `(0,1).` Then for every f in S, there exists a c `in (0,1)` depending on f, such that :

A

`|f(c)-f(1)|lt(1-c)|f'(c)|`

B

`(f(1)-f(c))/(1-c)=f'(c)`

C

`|f(c)-f(1)|lt|f'(c)|`

D

`|f(c)+f(1)|lt(1+c)|f'(c)|`

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Knowledge Check

  • Let the function, f:[-7,0]toR be continuous on [-7,0] and differentiable on (-7,0) . If f(-7)=-3 and f'(x)le2, for all x in(-7,0) , then for all such functions f,f(-1)+f(0) lies in the interval :

    A
    `[-6,20]`
    B
    `(-oo,20]`
    C
    `(-oo,11]`
    D
    `[-3,11]`

  • Let f be continuous on [0,1] and let F(x) be in [0,1] forall x in[0,1] forall x in[0,1] .Then,

    A
    `f(x)=x` for some `xepsilon[0.1]`
    B
    `f(x)=x^2,AAxepsilon[0,1]`
    C
    `f(x)=x-1,AAxepsilon[0,1]`
    D
    None of the above
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