Find the absolute maximum and absolute minimum values of \( f \) on the given interval. \[ f(x)=7+54 x-2 x^{3},[0,4] \] absolute minimum value absolute maximum value Blank \#1

To find the absolute maximum and minimum of the function \( f(x)=7+54x-2x^{3} \) on the interval \([0,4]\), we first need to evaluate the function at the endpoints and any critical points within the interval. 1. Calculate the derivative of \( f(x) \): \[ f'(x) = 54 - 6x^{2} \] Set the derivative equal to zero to find critical points: \[ 54 - 6x^{2} = 0 \implies 6x^{2} = 54 \implies x^{2} = 9 \implies x = 3 \] Since the interval is \([0, 4]\), \( x = 3 \) is within the bounds. 2. Evaluate \( f(x) \) at the critical point and the endpoints: \[ f(0) = 7 + 54(0) - 2(0)^{3} = 7 \] \[ f(3) = 7 + 54(3) - 2(3)^{3} = 7 + 162 - 54 = 115 \] \[ f(4) = 7 + 54(4) - 2(4)^{3} = 7 + 216 - 128 = 95 \] 3. Now, compare the values: - \( f(0) = 7 \) - \( f(3) = 115 \) - \( f(4) = 95 \) From these values, the absolute minimum value is \( 7 \) and the absolute maximum value is \( 115 \). Absolute minimum value: 7 Absolute maximum value: 115 Blank \#1: 115