find the absolute extremum, if any, for the following function.\n$f(x)=8x^{3}-1$\nselect the correct choice…
find the absolute extremum, if any, for the following function.\n$f(x)=8x^{3}-1$\nselect the correct choice below and, if necessary, fill in the answer boxes to complete your choice.\na. the absolute minimum is at $x=$\nb. there is no absolute minimum.
Answer
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\rightarrow-\infty), ( f(x)=8x^{3}-1\rightarrow-\infty). As ( x\rightarrow+\infty), ( f(x)=8x^{3}-1\rightarrow+\infty).
Answer:
B. There is no absolute minimum.