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For all twice differentiable functions...

For all twice differentiable functions ` f : R to R ` , with `f(0) = f(1) = f'(0) = 0`

A

`f(x) ne 0 ` at every point `x in (0,1)`

B

`f'(x) = 0` for some ` x ne (0,1)`

C

`f'(0) = 0`

D

`f''(x) = 0 ` , at every point ` x in (0,1)`

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Knowledge Check

  • Let f be a twice differentiable function defined on R such that f(0) = 1, f'(0) = 2 and f '(x) ne 0 for all x in R . If |[f(x)" "f'(x)], [f'(x)" "f''(x)]|= 0 , for all x in R , then the value of f(1) lies in the interval:

    A
    (9, 12)
    B
    (6, 9)
    C
    (0, 3)
    D
    (3,6)

  • For a differentiable function f on R there is an x_(0) with f'(x_(0))=(1)/(2)[f'(0)+f'(1)]

    A
    only if f is constant
    B
    only if f is increasing
    C
    if f is decreasing
    D
    if f is continuously differentiable

  • If f is twice differentiable function for x in R such that f(2)=5,f'(2)=8 and f'(x) ge 1,f''(x) ge4 , then

    A
    `f(5)+f'(5) le 26`
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    `f(5)+f'(5) ge 28`
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