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} )\n(type your answer in interval notation. use a comma to separate answers as needed.)\nwhat is the ( y )-intercept? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the ( y )-intercept is ( y = 0 ).\nb. there is no ( y )-intercept.\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 = ).\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 \nb. there is no local maximum.

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} )\n(type your answer in interval notation. use a comma to separate answers as needed.)\nwhat is the ( y )-intercept? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the ( y )-intercept is ( y = 0 ).\nb. there is no ( y )-intercept.\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 = ).\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 \nb. there is no local maximum.

Answer

Explanation:

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

The function is (f(x)=2x^{4}-12x^{2}). Using the power rule ((x^{n})^\prime = nx^{n - 1}), the derivative (f^\prime(x)=8x^{3}-24x=8x(x^{2}-3)=8x(x-\sqrt{3})(x + \sqrt{3}))

Step2: Find 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

The second - derivative (f^{\prime\prime}(x)=24x^{2}-24)

  • For (x = 0): (f^{\prime\prime}(0)=24\times0^{2}-24=-24<0). So (x = 0) is a local maximum.
  • For (x=\sqrt{3}): (f^{\prime\prime}(\sqrt{3})=24\times(\sqrt{3})^{2}-24=72 - 24 = 48>0). So (x=\sqrt{3}) is a local minimum.
  • For (x=-\sqrt{3}): (f^{\prime\prime}(-\sqrt{3})=24\times(-\sqrt{3})^{2}-24=72 - 24 = 48>0). So (x=-\sqrt{3}) is a local minimum.

Answer:

The local maximum is at (x = 0)