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

find the absolute extremum, if any, for the following function.\n\n$f(x)=3x^{4}-5$\n\nselect the correct choice below and, if necessary, fill in the answer boxes to complete your choice.\n\na. the absolute minimum is -5 at $x = 0$.\nb. there is no absolute minimum.\n\nselect the correct choice below and, if necessary, fill in the answer boxes to complete your choice.\n\na. the absolute maximum is at $x=$\nb. there is no absolute maximum.

find the absolute extremum, if any, for the following function.\n\n$f(x)=3x^{4}-5$\n\nselect the correct choice below and, if necessary, fill in the answer boxes to complete your choice.\n\na. the absolute minimum is -5 at $x = 0$.\nb. there is no absolute minimum.\n\nselect the correct choice below and, if necessary, fill in the answer boxes to complete your choice.\n\na. the absolute maximum is at $x=$\nb. there is no absolute maximum.

Answer

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: Analyze the second - derivative (or use the behavior of the function)

The second - derivative ( f^{\prime\prime}(x)=36x^{2} ). When ( x = 0 ), ( f^{\prime\prime}(0)=0 ). Another way: as ( x\rightarrow\pm\infty ), ( y = f(x)=3x^{4}-5\rightarrow+\infty ) since the leading term ( 3x^{4}) (where the degree ( n = 4) is even and the coefficient (a = 3>0)) dominates. And ( f(0)=3\times0^{4}-5=-5 ).

Answer:

A. The absolute minimum is (-5) at (x = 0); B. There is no absolute maximum.