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 (Use ascending order.) B. The absolute maximum is C. and \( x= \)

To find the absolute maximum and minimum of the function \( f(x) = x^4 + 4x^3 + 10 \), we need to evaluate it at critical points within the given interval and at the endpoints. First, let's find the derivative: \[ f'(x) = 4x^3 + 12x^2 \] Setting the derivative to zero gives us: \[ 4x^2(x + 3) = 0 \] This results in critical points at \( x = 0 \) and \( x = -3 \). Since we are examining the interval \([-2, 2]\), we will include \( x = -2 \), \( x = 0 \), and \( x = 2\). Next, let's evaluate the function at these critical points and the endpoints: 1. \( f(-2) = (-2)^4 + 4(-2)^3 + 10 = 16 - 32 + 10 = -6 \) 2. \( f(0) = 0^4 + 4(0)^3 + 10 = 10 \) 3. \( f(2) = 2^4 + 4(2)^3 + 10 = 16 + 32 + 10 = 58 \) Now we can see our results: - At \( x = -2 \), \( f(-2) = -6 \) - At \( x = 0 \), \( f(0) = 10 \) - At \( x = 2 \), \( f(2) = 58 \) The absolute maximum occurs at \( x = 2 \) with \( f(2) = 58 \). So, for option A: **The absolute maximum is** \( 58 \) **and** \( x = 2 \). Now, if you need further assistance, feel free to ask!