Find the absolute extremum, if any, for the following function \( f(x)=3 x^{4}-4 \) Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The absolute minimum is \( \square \) at \( x= \) B. There is no absolute minimum.

To find the absolute extremum of the function \( f(x) = 3x^4 - 4 \), we first need to determine the critical points by calculating the derivative and setting it to zero. The derivative of the function is: \[ f'(x) = 12x^3 \] Setting the derivative equal to zero gives: \[ 12x^3 = 0 \] This leads to: \[ x = 0 \] Next, we need to find the value of the function at this critical point: \[ f(0) = 3(0)^4 - 4 = -4 \] Now, we can evaluate the behavior of the function as \( x \) approaches positive and negative infinity: - As \( x \to \infty \), \( f(x) \to \infty \) - As \( x \to -\infty \), \( f(x) \to \infty \) Since the function goes to positive infinity in both directions and has a minimum at \( x=0 \), we conclude that the absolute minimum occurs at: A. The absolute minimum is \( -4 \) at \( x=0 \).