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. The absolute minimum is -4 at $$ x=0 $$. B. There is no absolute minimum Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. The absolute maximum is A. There is no absolute maximum B. A. A. A. A. A.

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Let the function be given by
$$
f(x)=3x^4-4.
$$

1. Find the derivative:
   $$
   f'(x)=12x^3.
   $$
   Setting $$ f'(x)=0 $$ gives
   $$
   12x^3=0 \quad \Rightarrow \quad x=0.
   $$

2. Evaluate the function at $$ x=0 $$:
   $$
   f(0)=3(0)^4-4=-4.
   $$
   Since $$ x=0 $$ is the only critical point, it is the candidate for the absolute minimum.

3. Behavior of the function:
   - Notice that $$ x^4 \geq 0 $$ for all $$ x $$ implies $$ 3x^4 \geq 0 $$ so that
     $$
     f(x)=3x^4-4\geq -4.
     $$
     The equality holds only at $$ x=0 $$, confirming the absolute minimum is $$-4$$ at $$ x=0 $$.
   - As $$ x $$ increases in magnitude (as $$ x\to\pm\infty $$), we have
     $$
     f(x)=3x^4-4 \to \infty.
     $$
     Therefore, there is no upper bound for $$ f(x) $$.

Thus:

- The absolute minimum is $$-4$$ at $$ x=0 $$.
- There is no absolute maximum.
The absolute minimum of the function $$ f(x)=3x^4-4 $$ is $$-4$$ at $$ x=0 $$. There is no absolute maximum for this function.

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