Let `f:[1/2,1]->R` (the set of all real numbers) be a positive, non-constant, and differentiable function such that `f^(prime)(x)<2f(x)a n df(1/2)=1` . Then the value of `int_(1/2)^1f(x)dx` lies in the interval (a)`(2e-1,2e)` (b) `(3-1,2e-1)` (c)`((e-1)/2,e-1)` (d) `(0,(e-1)/2)`

Options

A. `(2e-1,2e)`
B. `(e-1,2e-1)`
C. `((e-1)/2,e-1)`
D. `(0,(e-1)/2)`

Explanation

To solve the problem, we need to analyze the function \( f(x) \) under the given conditions and evaluate the integral \( \int_{1/2}^{1} f(x) \, dx \).

### Step-by-Step Solution:

1. **Understanding the Condition**: We have a function \( f: [1/2, 1] \to \mathbb{R} \) that is positive, non-constant, and differentiable. The condition given is \( f'(x) < 2f(x) \) for all \( x \in [1/2, 1] \) and \( f(1/2) = 1 \).

2. **Rearranging the Inequality**: We can rearrange the inequality \( f'(x) < 2f(x) \) to obtain:
\[
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