Step-by-step Solution:

Finding the Minimum Value of \( x + \frac{1}{x} \)
We need to minimize the function: \[ f(x) = x + \frac{1}{x} \] Step 1: Find the First Derivative
To find the critical points, differentiate \( f(x) \): \[ f'(x) = 1 - \frac{1}{x^2} \] Setting \( f'(x) = 0 \): \[ 1 - \frac{1}{x^2} = 0 \] \[ \frac{1}{x^2} = 1 \] \[ x^2 = 1 \] \[ x = \pm 1 \] Step 2: Second Derivative Test
Differentiate \( f'(x) \) again: \[ f''(x) = \frac{2}{x^3} \] Evaluate at \( x = 1 \): \[ f''(1) = \frac{2}{1^3} = 2 > 0 \] Since \( f''(1) > 0 \), \( x = 1 \) is a local minimum. Final Answer: \( {1} \)