Question

Find the absolute maximum and absolute minimum values of the function \[ f(x) = \frac{x^2}{2} + \frac{2}{x} \] on the interval \( [1, 2] \).

Hint:

To find absolute extrema, evaluate \( f(x) \) at critical points and endpoints within the interval.

Answer 1

1. Derivative calculation: \( f'(x) = \frac{1}{2} - \frac{2}{x^2} \).
2. Critical points identification: Setting \( f'(x) = 0 \) yields \( \frac{1}{2} - \frac{2}{x^2} = 0 \), which simplifies to \( \frac{2}{x^2} = \frac{1}{2} \), then \( x^2 = 4 \), and finally \( x = 2 \). Only \( x = 2 \) is within the interval \( [1, 2] \).
3. Function evaluation at endpoints and critical point: - At \( x = 1 \): \( f(1) = \frac{1}{2} + \frac{2}{1} = \frac{1}{2} + 2 = 2.5 \). - At \( x = 2 \): \( f(2) = \frac{2}{2} + \frac{2}{2} = 1 + 1 = 2 \).
4. Determining absolute extrema: The absolute maximum value is \( 2.5 \) occurring at \( x = 1 \), and the absolute minimum value is \( 2 \) occurring at \( x = 2 \).
Final Answer: \( \text{Absolute maximum: } 2.5 \, (\text{at } x = 1), \quad \text{Absolute minimum: } 2 \, (\text{at } x = 2). \)