Consider the function ` f(x)=x+(1)/(x) , x in((1)/(2),(9)/(2))` . If `alpha` is the length of interval of decreasing and `beta` be the length of internal of increase , then ` (beta)/(alpha)` is ____________

Explanation

To solve the given problem, we need to analyze the function \( f(x) = x + \frac{1}{x} \) over the interval \( \left( \frac{1}{2}, \frac{9}{2} \right) \) and determine where it is increasing and decreasing. We will then find the lengths of these intervals and compute the ratio \( \frac{\beta}{\alpha} \).

### Step-by-Step Solution:

1. **Differentiate the Function:**
We start by finding the derivative of the function \( f(x) \):
\[
f'(x) = \frac{d}{dx}\left(x + \frac{1}{x}\right) = 1 - \frac{1}{x^2}
...