Question

Let f:[1/2, 1] →R(the set all real numbers) be a positive, non-constant and differentiable function such that f'(x) < 2f(x) and f(1/2) = 1. Then the value of ∫f(x)dx for x ∈ [1/2, 1] the interval

(A) (2e - 1, 2e)

(B) (e - 1, 2e - 1)

(C) ((e - 1)/2, e - 1)

(D) (0, (e - 1)/2)

Answer 1

Answer is (D) (0, (e - 1)/2)

We have

which implies that ye−2x is a decreasing function.