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.

Explanation:

Step1: Find the derivative of the function

The derivative of \(f(x)=3x^{3}-7\) is \(f^\prime(x) = 9x^{2}\) using the power rule \((x^{n})^\prime=nx^{n - 1}\).

Step2: Find the critical points

Set \(f^\prime(x)=0\), so \(9x^{2}=0\). Solving for \(x\), we get \(x = 0\).

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

As \(x\to\infty\), \(y = f(x)=3x^{3}-7\to\infty\) (since the leading term \(3x^{3}\) dominates).
As \(x\to-\infty\), \(y = f(x)=3x^{3}-7\to-\infty\) (since the leading term \(3x^{3}\) dominates).

Answer:

B. There is no absolute minimum.