Question

find the absolute extremum, if any, for the following function.
$f(x)=8x^{3}-1$
select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
a. the absolute minimum is at $x=$
b. there is no absolute minimum.

Explanation:

Step1: Find the derivative

The derivative of \( f(x)=8x^{3}-1\) is \( f^\prime(x) = 24x^{2}\).

Step2: Set the derivative equal to zero

Set \( f^\prime(x)=0\), so \( 24x^{2}=0\). Solving for \( x\), we get \( x = 0\).

Step3: Analyze the behavior of the function

As \( x
ightarrow-\infty\), \( f(x)=8x^{3}-1
ightarrow-\infty\). As \( x
ightarrow+\infty\), \( f(x)=8x^{3}-1
ightarrow+\infty\).

Answer:

B. There is no absolute minimum.