determine all critical points for the following function.\nf(x)=2x(16 - x)^3\nwhat is the derivative of…

determine all critical points for the following function.\nf(x)=2x(16 - x)^3\nwhat is the derivative of f(x)=2x(16 - x)^3?\nf(x)=8(16 - x)^2(4 - x)\nfind the critical points, if any, of f on the domain. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the critical point(s) occur(s) at x=\n(use a comma to separate answers as needed.)\nb. there are no critical points for f(x)=2x(16 - x)^3 on the domain.

determine all critical points for the following function.\nf(x)=2x(16 - x)^3\nwhat is the derivative of f(x)=2x(16 - x)^3?\nf(x)=8(16 - x)^2(4 - x)\nfind the critical points, if any, of f on the domain. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the critical point(s) occur(s) at x=\n(use a comma to separate answers as needed.)\nb. there are no critical points for f(x)=2x(16 - x)^3 on the domain.

Answer

Explanation:

Step1: Recall the definition of critical points

Critical points occur where (f^{\prime}(x) = 0) or (f^{\prime}(x)) is undefined. Since (f^{\prime}(x)=8(16 - x)^{2}(4 - x)) is a polynomial, it is defined for all real (x). So we only need to solve (f^{\prime}(x)=0).

Step2: Set (f^{\prime}(x)=0)

Set (8(16 - x)^{2}(4 - x)=0). Using the zero - product property (ab = 0) implies (a = 0) or (b = 0). If (8(16 - x)^{2}(4 - x)=0), then either ((16 - x)^{2}=0) or ((4 - x)=0). For ((16 - x)^{2}=0), we solve (16 - x=0), which gives (x = 16). For (4 - x=0), we solve (x = 4).

Answer:

A. The critical point(s) occur(s) at (x = 4,16)