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`

Options

A. `f'(x)ltf(x),(1)/(4)ltxlt(3)/(4)`
B. `f'(x)gtf(x),0ltxlt(1)/(4)`
C. `f'(x)ltf(x),0ltxlt(1)/(4)`
D. `f'(x)ltf(x),(3)/(4)ltxlt1`

Answer

C