Let α, β ∈ &bold;R&bold; be such that the function: f(x) = 2α(x2 - 2) + 2βx, for x < 1 (α + 3)x + (α - β), for x ≥ 1 is differentiable at all x ∈ &bold;R&bold;. Then 34(α + β) is equal to:

Let a function f : (-π/4, π/4) → R be defined as: f(x) = (1 + |sin x|)3a/|sin x|, if -π/4 < x < 0 f(x) = b, if x = 0 f(x) = ecot 4x / cot 2x, if 0 < x < π/4 If f is continuous at x = 0, then the value of 6a + b2 is equal to:

Let f : [0, ∞) → [0, ∞) be defined as f(x) = ∫0x [y] dy, where [y] is the greatest integer less than or equal to y. Which of the following is true?