find the critical points of the following function.\n\n$f(x)=\\frac{4x}{x^{2}+16}$\n\nwhat is the derivative…

find the critical points of the following function.\n\n$f(x)=\\frac{4x}{x^{2}+16}$\n\nwhat is the derivative of $f(x)=\\frac{4x}{x^{2}+16}$?\n\n$f(x)=\\frac{-4x^{2}+64}{(x^{2}+16)^{2}}$\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the critical point(s) occur(s) at $x=$\n(use a comma to separate answers as needed.)\n\nb. there are no critical points.

find the critical points of the following function.\n\n$f(x)=\\frac{4x}{x^{2}+16}$\n\nwhat is the derivative of $f(x)=\\frac{4x}{x^{2}+16}$?\n\n$f(x)=\\frac{-4x^{2}+64}{(x^{2}+16)^{2}}$\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the critical point(s) occur(s) at $x=$\n(use a comma to separate answers as needed.)\n\nb. there are no critical points.

Answer

Explanation:

Step1: Set the derivative equal to zero

To find critical points, set (f^{\prime}(x)=0). So, (\frac{-4x^{2}+64}{(x^{2}+16)^{2}} = 0). Since the denominator ((x^{2}+16)^{2}>0) for all real (x) (because (x^{2}\geq0), so (x^{2}+16>0)), we only need to solve (-4x^{2}+64 = 0).

Step2: Solve the equation (-4x^{2}+64 = 0)

First, rewrite the equation as (4x^{2}=64). Then (x^{2}=\frac{64}{4}=16). Taking square roots, (x=\pm4).

Answer:

A. The critical point(s) occur(s) at (x = - 4,4)