find the absolute extremum, if any, for the following function.\n\n( f(x)=3 x^{4}-5 )\n\nselect the correct…

Explanation:

Step1: Find the derivative of the function

The function is (f(x) = 3x^{4}-5). Using the power rule ((x^{n})^\prime=nx^{n - 1}), the derivative (f^\prime(x)=3\times4x^{3}=12x^{3}).

Step2: Find the critical points

Set (f^\prime(x) = 0), so (12x^{3}=0). Solving for (x), we get (x = 0).

Step3: Use the second - derivative test

Find the second - derivative (f^{\prime\prime}(x)). Differentiating (f^\prime(x)=12x^{3}) with respect to (x) using the power rule, (f^{\prime\prime}(x)=36x^{2}). Evaluate (f^{\prime\prime}(x)) at (x = 0): (f^{\prime\prime}(0)=36\times0^{2}=0). We can also use the behavior of the function. As (x\rightarrow\pm\infty), (y = f(x)=3x^{4}-5\rightarrow+\infty) (since the leading term (3x^{4}) dominates). Substitute (x = 0) into the function (f(x)): (f(0)=3\times0^{4}-5=- 5).

Answer:

A. The absolute minimum is (-5) at (x = 0).