Mathematics

Problem

Let

f: \left[\frac{1}{2}, 1\right] \to \mathbb{R}

(the set of all real numbers) be a positive, non-constant and differentiable function such that

f'(x) < 2f(x)

and

f\left(\frac{1}{2}\right) = 1

. Then the value of

\int_{1/2}^{1} f(x)\,dx

lies in the interval [JEE(Adv.) 2013] (A)

(2e - 1, 2e)

(B)

(e - 1, 2e - 1)

(C)

\left(\frac{e-1}{2}, e-1\right)

(D)

\left(0, \frac{e-1}{2}\right)