### Problem Statement 11. (10 points) Find the absolute extrema of the function \( f(x) = \frac{1}{6}x^2 - 10x + 200 \) on the interval \([0, 48]\). Give the extreme values and the inputs at which they occur. ### Questions - \( f(x) \) has an absolute **maximum** of \_\_\_\_\_\_\_\_\_ at \( x = \_\_\_\_\_\_\_\_\_ \). - \( f(x) \) has an absolute **minimum** of \_\_\_\_\_\_\_\_\_ at \( x = \_\_\_\_\_\_\_\_\_ \). ### Instructions 1. **Determine Critical Points:** - Find the derivative of \( f(x) \). - Solve \( f'(x) = 0 \) to find critical points. 2. **Evaluate Extreme Values:** - Calculate \( f(x) \) at the critical points and the endpoints \( x = 0 \) and \( x = 48 \). 3. **Identify Absolute Extrema:** - Compare the values from the previous step to determine the absolute maximum and minimum. 4. **Complete the Statements:** - Fill in the blanks with the correct values and \( x \)-coordinates.