Let f : [0, 1]→R be a function. Suppose the function f is twice differentiable f(0) = 0 = f(1) and satisfies f''(x) – 2f'(x) + f(x) ≥ ex, x∈[0, 1]
(i) which of the following is true for 0 < x < 1?
(a) 0 < f(x) < ∞
(b) – 1/2 < f(x) < 1/2
(c) – 1/4 < f(x) < 1
(d) – ∞ < f(x) < 0
(ii) 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) < f(x), 1/4 < x < 3/4
(b) f(x) > f(x), 0 < x < 1/4
(c) f'(x) < f(x), 0 < x < 1/4
(d) f'(x) < f(x), 3/4 < x < 1
