Question

Find the absolute maximum and absolute minimum values of the function \( f(x) = \frac{x}{2} + \frac{2}{x} \) 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. Identification of Critical Points: Setting \( f'(x) = 0 \) yields \( \frac{1}{2} - \frac{2}{x^2} = 0 \), which simplifies to \( \frac{2}{x^2} = \frac{1}{2} \) and further to \( x^2 = 4 \). The solutions are \( x = 2 \). Within the interval \( [1, 2] \), only \( x = 2 \) is considered. 3. Evaluation at Boundary and Critical Points: - For \( x = 1 \): \( f(1) = \frac{1}{2} + \frac{2}{1} = \frac{1}{2} + 2 = 2.5. \) - For \( x = 2 \): \( f(2) = \frac{2}{2} + \frac{2}{2} = 1 + 1 = 2. \) 4. Determination of Extrema: The absolute maximum value is \( 2.5 \) occurring at \( x = 1 \). The absolute minimum value is \( 2 \) occurring at \( x = 2 \). Final Answer: \( {Absolute maximum: } 2.5 \, ({at } x = 1), \quad {Absolute minimum: } 2 \, ({at } x = 2). \)