find the x-values of all points where the function has any relative extrema. find the value(s) of any…

find the x-values of all points where the function has any relative extrema. find the value(s) of any relative extrema.\nf(x)=x^{4}-128x^{2}+4096\nselect the correct choice below and, if necessary, fill in any answer boxes within your choice.\na. the function has a relative maximum of \nand a relative minimum of \nat x=\nat x=\n(use a comma to separate answers as needed.)\nb. there are no relative maxima. the function has a relative minimum of \nat x=\n(use a comma to separate answers as needed.)\nc. there are no relative minima. the function has a relative maximum of \nat x=\n(use a comma to separate answers as needed.)\nd. there are no relative extrema.

find the x-values of all points where the function has any relative extrema. find the value(s) of any relative extrema.\nf(x)=x^{4}-128x^{2}+4096\nselect the correct choice below and, if necessary, fill in any answer boxes within your choice.\na. the function has a relative maximum of \nand a relative minimum of \nat x=\nat x=\n(use a comma to separate answers as needed.)\nb. there are no relative maxima. the function has a relative minimum of \nat x=\n(use a comma to separate answers as needed.)\nc. there are no relative minima. the function has a relative maximum of \nat x=\n(use a comma to separate answers as needed.)\nd. there are no relative extrema.

Answer

Explanation:

Step1: Find the first derivative

Given ( f(x)=x^{4}-128x^{2}+4096 ). Using the power rule ( (x^n)^\prime = nx^{n - 1} ), we have ( f^\prime(x)=4x^{3}-256x=4x(x^{2}-64)=4x(x - 8)(x + 8) ).

Step2: Find the critical points

Set ( f^\prime(x)=0 ). ( 4x(x - 8)(x + 8)=0 ). By the zero - product property ( x=0) or (x = 8) or (x=-8).

Step3: Find the second derivative

Using the power rule again, ( f^{\prime\prime}(x)=12x^{2}-256 ).

Step4: Use the second - derivative test

  • For ( x = 0): ( f^{\prime\prime}(0)=12\times0^{2}-256=-256<0 ). So ( f(x)) has a relative maximum at ( x = 0).
  • For ( x = 8): ( f^{\prime\prime}(8)=12\times8^{2}-256=12\times64 - 256=768-256 = 512>0 ). So ( f(x)) has a relative minimum at ( x = 8).
  • For ( x=-8): ( f^{\prime\prime}(-8)=12\times(-8)^{2}-256=12\times64 - 256=768 - 256=512>0 ). So ( f(x)) has a relative minimum at ( x=-8).

Answer:

B. There are no relative maxima. The function has a relative minimum of ( f(-8)=(-8)^{4}-128\times(-8)^{2}+4096=4096-8192 + 4096 = 0), ( f(8)=8^{4}-128\times8^{2}+4096=4096-8192 + 4096=0) at (x=-8) and (x = 8).