If the function e⁻ˣf(x) assumes its minimum in the interval [0,1] at x=1/4, which of the following is true?

Define a function g(x) = e⁻ˣf(x) ⇒g'(x) = e⁻ˣ(f'(x) - f(x)) ⇒g''(x) = e⁻ˣ[f''(x) - f'(x) + f(x)]
Given that f''(x) - f'(x) + f(x) ≥ eˣ, x ∈ [0,1] ⇒e⁻ˣ(f''(x) - f'(x) + f(x)) ≥ 1
Hence g''(x) ≥ 1 > 0
So g'(x) is increasing.
So, for x < 1/4, g'(x) < g'(1/4) ⇒g'(x) < 0 ⇒(f'(x) - f(x))e⁻ˣ < 0 ⇒f'(x) < f(x) in (0,1/4).