find the absolute extremum, if any, for the following function. f(x)=3x^4 - 7 select the correct choice…

Explanation:

Step1: Find the derivative

Differentiate $f(x)=3x^{4}-7$ using the power - rule. The derivative $f^\prime(x)=12x^{3}$.

Step2: Find critical points

Set $f^\prime(x) = 0$. So, $12x^{3}=0$, which gives $x = 0$.

Step3: Analyze the second - derivative

Differentiate $f^\prime(x)$ to get the second - derivative $f^{\prime\prime}(x)=36x^{2}$. Evaluate $f^{\prime\prime}(0)=0$. But we can also analyze the function behavior. Since $x^{4}\geq0$ for all real $x$, then $3x^{4}\geq0$ for all real $x$, and $f(x)=3x^{4}-7\geq - 7$ for all real $x$. So the absolute minimum is $-7$ at $x = 0$.

Step4: Analyze the end - behavior

As $x\rightarrow\pm\infty$, $y = f(x)=3x^{4}-7\rightarrow+\infty$. So there is no absolute maximum.

Answer:

A. The absolute minimum is $-7$ at $x = 0$. B. There is no absolute maximum.