Question

Let f; [0, ∞) → R be a continuous function such that f(x) = 1 - 2x +  ∫e^{x - t}f(t)dt for t ∈ [0, x] for all x ∈ [0, ∞). Then, which of the following statement(s) is(are) TRUE? 

(A) The curve y = f(x) passes through the point (1, 2). 

(B) The curve y = f(x) passes through the point (2, –1). 

(C) The area of the region {(x, y)

Answer 1

Answer is (B), (C)

We have

Multiplying both sides by e^−x, we get

Differentiating, we get

Here, the integrating factor is e^−2x. Therefore,

Using integration by parts, we get

These lines pass through (2, −1).

Hence, option (B) is correct.