summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y =…

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y = f(x) ).\n( f(x)=2 x^{4}-12 x^{2} )\nwhat is/are the ( x )-intercept(s)? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the ( x )-intercept(s) is/are ( x=-sqrt{6}, 0, sqrt{6} ).\n(type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\nb. there is no ( x )-intercept.\nwhat is/are the local maximum/a? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the local maximum/a is/are at ( x = 0 ).\n(simplify your answer. type an exact answer using radicals as needed. use integers or fractions for any numbers in the expression. use a comma to sepa\nb. there is no local maximum.\nwhat is/are the local minimum/a? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the local minimum/a is/are at ( x=square ).\n(simplify your answer type an exact answer using radicals as needed. use a comma to separate answers as needed.)\nb. there is no local minimum.

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y = f(x) ).\n( f(x)=2 x^{4}-12 x^{2} )\nwhat is/are the ( x )-intercept(s)? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the ( x )-intercept(s) is/are ( x=-sqrt{6}, 0, sqrt{6} ).\n(type an exact answer, using radicals as needed. use a comma to separate answers as needed.)\nb. there is no ( x )-intercept.\nwhat is/are the local maximum/a? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the local maximum/a is/are at ( x = 0 ).\n(simplify your answer. type an exact answer using radicals as needed. use integers or fractions for any numbers in the expression. use a comma to sepa\nb. there is no local maximum.\nwhat is/are the local minimum/a? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the local minimum/a is/are at ( x=square ).\n(simplify your answer type an exact answer using radicals as needed. use a comma to separate answers as needed.)\nb. there is no local minimum.

Answer

Explanation:

Step1: Find the derivative of (f(x))

Given (f(x)=2x^{4}-12x^{2}), then (f^{\prime}(x)=8x^{3}-24x = 8x(x^{2}-3)=8x(x - \sqrt{3})(x+\sqrt{3}))

Step2: Find the critical points

Set (f^{\prime}(x)=0), so (8x(x - \sqrt{3})(x+\sqrt{3})=0). The critical points are (x = 0,x=\sqrt{3},x =-\sqrt{3})

Step3: Use the second - derivative test

First, find the second - derivative (f^{\prime\prime}(x)=24x^{2}-24)

  • When (x = 0), (f^{\prime\prime}(0)=24\times0^{2}-24=- 24<0), so (x = 0) is a local maximum.
  • When (x=\sqrt{3}), (f^{\prime\prime}(\sqrt{3})=24\times(\sqrt{3})^{2}-24=24\times3 - 24=48>0), so (x=\sqrt{3}) is a local minimum.
  • When (x=-\sqrt{3}), (f^{\prime\prime}(-\sqrt{3})=24\times(-\sqrt{3})^{2}-24=24\times3 - 24=48>0), so (x=-\sqrt{3}) is a local minimum.

Answer:

The local minimum/a is/are at (x=-\sqrt{3},\sqrt{3})