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Let f:[0,1]rarrR (the set of all real nu...

Let `f:[0,1]rarrR` (the set of all real numbers) be a function. Suppose the function `f` is twice differentiable, `f(0)=f(1)=0 and satisfies `f\'\'(x)-2f\'(x)+f(x) ge e^x, x in [0,1]` If the function `e^(-x)f(x)` assumes its minimum in the interval `[0,1]` at `x=1/4`, which of the following is true? (A) `f\'(x) lt f(x), 1/4 lt x lt 3/4` (B) `f\'(x) gt f(x), `0 lt x lt 1/4` (C) `f\'(x) lt f(x), 0 lt x lt 1/4` (D) `f\'(x) lt f(x), 3/4 lt x lt 1`

A

`f' (x) lt f(x), (1)/(4) lt x lt (3)/(4)`

B

`f' (x) gt f(x), 0 lt x lt (1)/(4)`

C

`f' (x) lt f(x), 0 lt x lt (1)/(4)`

D

`f' (x) lt f(x), (3)/(4) lt x lt 1`

Text Solution

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The correct Answer is:
C
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