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) le f(x)`
B. `f'(x) ge f(x) ,0 le x le (1)/(4)`
C. ` f'(x) le f(x) , 0 le x le (1)/(4)`
D. ` f'(x) le f(x) , (3)/(4) le x le 1`

Answer

C