Find the absolute extremum, if any, for the following function. \( f(x)=7 x^{4}-4 \) Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The absolute minimum is \( \square \) at \( x=\square \). B. There is no absolute minimum. Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The absolute maximum is \( \square \) at \( x=\square \). B. There is no absolute maximum.

To find the absolute extrema of the function \( f(x) = 7x^4 - 4 \), we first need to analyze this polynomial function. Since it is a fourth-degree polynomial with a positive leading coefficient, it opens upwards. To find critical points, we take the derivative of the function: \[ f'(x) = 28x^3 \] Setting the derivative equal to zero: \[ 28x^3 = 0 \] This gives us: \[ x = 0 \] Now, we evaluate the original function at this critical point: \[ f(0) = 7(0)^4 - 4 = -4 \] Since \( f(x) \to \infty \) as \( x \to \pm\infty \), there are no upper bounds on \( f(x) \), meaning there is no absolute maximum. The lowest point, or absolute minimum, occurs at the critical point we found. Thus, the answers are: A. The absolute minimum is \(-4\) at \( x=0 \). B. There is no absolute maximum.