\( f(x)=\sqrt [3]{x};[1,27] \)

To analyze the function \( f(x) = \sqrt[3]{x} \) over the interval \([1, 27]\), we can follow these steps: 1. **Determine the values of the function at the endpoints of the interval**: - At \( x = 1 \): \[ f(1) = \sqrt[3]{1} = 1 \] - At \( x = 27 \): \[ f(27) = \sqrt[3]{27} = 3 \] 2. **Check the behavior of the function within the interval**: - The function \( f(x) = \sqrt[3]{x} \) is a continuous and increasing function for all \( x \). This means that as \( x \) increases from 1 to 27, \( f(x) \) will also increase. 3. **Find the minimum and maximum values**: - Since \( f(x) \) is increasing on the interval \([1, 27]\): - The minimum value occurs at \( x = 1 \): \( f(1) = 1 \) - The maximum value occurs at \( x = 27 \): \( f(27) = 3 \) 4. **Conclusion**: - The minimum value of \( f(x) \) on the interval \([1, 27]\) is \( 1 \). - The maximum value of \( f(x) \) on the interval \([1, 27]\) is \( 3 \). Thus, the function \( f(x) = \sqrt[3]{x} \) on the interval \([1, 27]\) has a minimum value of \( 1 \) and a maximum value of \( 3 \).