Find the absolute maximum and minimum, if either exists, for the function on the indicated interval \( f(x)=x^{4}+4 x^{3}+10 \) \( \begin{array}{lll}\text { (A) }[-2,2] & \text { (B) }[-4,0] & \text { (C) }[-2,1]\end{array} \) (A) Find the absolute maximum. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The absolute maximum, which occurs twice, is at \( x= \) and \( x= \) (Use ascending order.) B. The absolute maximum is 58 at \( x=2 \) C. There is no absolute maximum. Find the absolute minimum. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The absolute minimum, which occurs twice, is \( ] \) at \( x=\square \) and \( x=\square \) (Use ascending order.) B. The absolute minimum is \( \square \) at \( x=\square \) C. There is no absolute minimum.

To find the absolute maximum and minimum of the function \( f(x) = x^4 + 4x^3 + 10 \) on the interval \([-2, 2]\), we first evaluate the function at the endpoints of the interval and any critical points. 1. **Endpoints**: - \( f(-2) = (-2)^4 + 4(-2)^3 + 10 = 16 - 32 + 10 = -6 \) - \( f(2) = 2^4 + 4(2)^3 + 10 = 16 + 32 + 10 = 58 \) 2. **Critical Points**: - Find \( f'(x) \): \[ f'(x) = 4x^3 + 12x^2 \] Set \( f'(x) = 0 \): \[ 4x^2(x + 3) = 0 \] This gives us critical points at \( x = 0 \) and \( x = -3 \). Since \(-3\) is outside the interval, we only consider \( x = 0 \). - Calculate \( f(0) \): \[ f(0) = 0^4 + 4(0)^3 + 10 = 10 \] 3. **Comparing Values**: - \( f(-2) = -6 \) - \( f(0) = 10 \) - \( f(2) = 58 \) Now let's summarize the results: - The absolute maximum is \( 58 \) at \( x = 2 \). - The absolute minimum is \( -6 \) at \( x = -2 \). Answer selection: (A) The absolute maximum is 58 at \( x=2 \). (B) The absolute minimum is \(-6\) at \( x=-2 \).