\( f ( x ) = x ^ { 4 } \quad [ 1,3 ] \)

To analyze the function \( f(x) = x^4 \) on the interval \([1, 3]\), we can perform several steps, including finding the function's values at the endpoints, determining critical points, and evaluating the function at those points. ### Step 1: Evaluate the function at the endpoints Calculate \( f(1) \) and \( f(3) \): \[ f(1) = 1^4 = 1 \] \[ f(3) = 3^4 = 81 \] ### Step 2: Find critical points To find critical points, we first need to compute the derivative of \( f(x) \): \[ f'(x) = 4x^3 \] Next, we set the derivative equal to zero to find critical points: \[ 4x^3 = 0 \implies x^3 = 0 \implies x = 0 \] However, \( x = 0 \) is not in the interval \([1, 3]\). Therefore, there are no critical points in this interval. ### Step 3: Evaluate the function at the endpoints Since there are no critical points in the interval, we only need to evaluate the function at the endpoints: - \( f(1) = 1 \) - \( f(3) = 81 \) ### Step 4: Determine the minimum and maximum values From our evaluations: - The minimum value of \( f(x) \) on the interval \([1, 3]\) is \( f(1) = 1 \). - The maximum value of \( f(x) \) on the interval \([1, 3]\) is \( f(3) = 81 \). ### Conclusion The minimum value of \( f(x) = x^4 \) on the interval \([1, 3]\) is \( 1 \) at \( x = 1 \), and the maximum value is \( 81 \) at \( x = 3 \).