Question

find the absolute extremum, if any, for the following function.
f(x)=3x^{3}-7
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.
select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
a. the absolute maximum is at x=
b. there is no absolute maximum.

Explanation:

Step1: Find the derivative

The derivative of \(f(x)=3x^{3}-7\) is \(f^\prime(x)=9x^{2}\).

Step2: Find critical points

Set \(f^\prime(x) = 0\), so \(9x^{2}=0\), which gives \(x = 0\).

Step3: Analyze the behavior of the function as \(x\to\pm\infty\)

As \(x\to\infty\), \(f(x)=3x^{3}-7\to\infty\).
As \(x\to-\infty\), \(f(x)=3x^{3}-7\to-\infty\).

Answer:

B. There is no absolute minimum.
B. There is no absolute maximum.