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.

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UpStudy AI · Accepted Answer

To find the absolute extremum of the function $$ f(x) = 7x^4 - 4 $$, we need to determine if there are any critical points where the function changes from increasing to decreasing or vice versa.

To find the critical points, we take the derivative of the function with respect to $$ x $$ and set it equal to zero:

$$ f'(x) = 28x^3 = 0 $$

Solving for $$ x $$, we find that the critical point is at $$ x = 0 $$.

Now, we need to determine the nature of the critical point by examining the second derivative:

$$ f''(x) = 84x^2 $$

At $$ x = 0 $$, the second derivative is zero, which indicates that the critical point is neither a local maximum nor a local minimum. Therefore, there is no absolute extremum for the function $$ f(x) = 7x^4 - 4 $$.

So, the correct choices are:
A. There is no absolute minimum.
B. There is no absolute maximum.
There is no absolute minimum or maximum for the function $$ f(x) = 7x^4 - 4 $$.

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