Find the absolute maximum and minimum, if either exists, \( f \) or \( f(x)=x+\frac{25}{x} \). Find the second derivative of \( f \). \( f^{\prime \prime}(x)=\square \) Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The absolute maximum is Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The absolute minimum is B. There is no absolute minimum. B.

To find the absolute maximum and minimum of \( f(x) = x + \frac{25}{x} \), we first need to find the first derivative: \[ f'(x) = 1 - \frac{25}{x^2}. \] Next, we set \( f'(x) = 0 \) to find critical points: \[ 1 - \frac{25}{x^2} = 0 \] \[ \frac{25}{x^2} = 1 \] \[ x^2 = 25 \] \[ x = 5 \quad \text{(since } x > 0\text{ for the domain of } f). \] Now, we'll check the second derivative to determine concavity: \[ f''(x) = \frac{50}{x^3}. \] Evaluating \( f''(5) \): \[ f''(5) = \frac{50}{5^3} = \frac{50}{125} = \frac{2}{5}, \] which is positive, indicating that \( x = 5 \) is a local minimum. Now, we evaluate \( f \) at the critical point: \[ f(5) = 5 + \frac{25}{5} = 5 + 5 = 10. \] As \( x \) approaches \( 0^+ \), \( f(x) \to +\infty \) and as \( x \to \infty \), \( f(x) \to +\infty \) as well, suggesting no absolute maximum exists within the interval. **Final answers:** \( f^{\prime \prime}(x) = \frac{50}{x^3} \) A. The absolute maximum is none. A. The absolute minimum is 10.