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: Determine the nature of the function

Take the second - derivative $f^{\prime\prime}(x)=36x^{2}$. Evaluate $f^{\prime\prime}(0)=0$. We can also analyze the behavior of the function. Since the leading coefficient of $f(x)=3x^{4}-7$ is positive ($a = 3>0$) and the degree is even ($n = 4$), the function opens upwards.

Step4: Find the absolute minimum

Evaluate $f(x)$ at the critical point $x = 0$. $f(0)=3\times0^{4}-7=-7$.

Answer:

A. The absolute minimum is $-7$ at $x = 0$.