Let 𝑓(𝑥) be a continuous function from ℝ to ℝ such that
𝑓(𝑥) = 1 − 𝑓(2 − 𝑥)
Which one of the following options is the CORRECT value of $∫_0^2 𝑓(𝑥)𝑑𝑥$?

Question

Let 𝑓(𝑥) be a continuous function from ℝ to ℝ such that 𝑓(𝑥) = 1 − 𝑓(2 − 𝑥) Which one of the following options is the CORRECT value of $∫_0^2 𝑓(𝑥)𝑑𝑥$?

Accepted Answer

Correct answer: B — Key property: f(x) + f(2 − x) = 1 for all real x. Let I = ∫_0^2 f(x) dx. Integrate the given relation over [0,2]: ∫_0^2 f(x) dx + ∫_0^2 f(2 − x) dx = ∫_0^2 1 dx = 2. Change variables in the second integral with u = 2 − x. Then ∫_0^2 f(2 − x) dx = ∫_2^0 f(u) (−du) = ∫_0^2 f(u) du = I. Therefore I + I = 2, so 2I = 2 and I = 1. Conclusion: ∫_0^2 f(x) dx = 1.

Answer

A. 0

Answer

B. 1

Answer

C. 2

Answer

D. −1

Let 𝑓(𝑥) be a continuous function from ℝ to ℝ such that 𝑓(𝑥) = 1 − 𝑓(2 −…