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} )\na. the local minimum/a is/are at ( x=-sqrt{3}, sqrt{3} )\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.\nwhat are the inflection points? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the inflection points are at ( x=-1,1 )\n(simplify your answer. type an exact answer using radicals as needed. use a comma to separate answers as needed)\nb. there are no inflection points.\non what interval(s) is ( f ) increasing or decreasing?\n(type your answer in interval notation. use a comma to separate answers as needed. use integers or fractions for any numbers in the expression.)\na. ( f ) is increasing on ( square ) and ( f ) is decreasing on ( square )\nb. ( f ) is never increasing: ( f ) is decreasing on ( square )\nc. ( f ) is never decreasing. ( f ) is increasing on ( square )

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} )\na. the local minimum/a is/are at ( x=-sqrt{3}, sqrt{3} )\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.\nwhat are the inflection points? select the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. the inflection points are at ( x=-1,1 )\n(simplify your answer. type an exact answer using radicals as needed. use a comma to separate answers as needed)\nb. there are no inflection points.\non what interval(s) is ( f ) increasing or decreasing?\n(type your answer in interval notation. use a comma to separate answers as needed. use integers or fractions for any numbers in the expression.)\na. ( f ) is increasing on ( square ) and ( f ) is decreasing on ( square )\nb. ( f ) is never increasing: ( f ) is decreasing on ( square )\nc. ( f ) is never decreasing. ( f ) is increasing on ( square )

Answer

Explanation:

Step1: Find the first derivative

Using the power rule ((x^n)^\prime=nx^{n - 1}), for (y = f(x)=2x^{4}-12x^{2}), the first 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), then (8x(x-\sqrt{3})(x+\sqrt{3})=0). The critical points are (x = 0,x=\sqrt{3},x=-\sqrt{3})

Step3: Use the first - derivative test

  • For (x<-\sqrt{3}), let (x=-2), then (f^\prime(-2)=8\times(-2)\times((-2)^{2}-3)=-16<0)
  • For (-\sqrt{3}<x<0), let (x = - 1), then (f^\prime(-1)=8\times(-1)\times((-1)^{2}-3)=16>0)
  • For (0<x<\sqrt{3}), let (x = 1), then (f^\prime(1)=8\times1\times(1^{2}-3)=-16<0)
  • For (x>\sqrt{3}), let (x = 2), then (f^\prime(2)=8\times2\times(2^{2}-3)=16>0)

Since (f(x)) changes from decreasing ((x<-\sqrt{3})) to increasing ((-\sqrt{3}<x<0)) at (x =-\sqrt{3}) and from decreasing ((0<x<\sqrt{3})) to increasing ((x>\sqrt{3})) at (x=\sqrt{3}), the local minima are at (x=-\sqrt{3},\sqrt{3})

Step4: Find the second derivative

(f^\prime(x)=8x^{3}-24x), then (f^{\prime\prime}(x)=24x^{2}-24 = 24(x^{2}-1)=24(x - 1)(x + 1))

Step5: Find inflection points

Set (f^{\prime\prime}(x)=0), then (24(x - 1)(x + 1)=0), so (x=-1,x = 1)

Step6: Determine increasing and decreasing intervals

From the first - derivative test: The function (f(x)) is increasing on ((-\sqrt{3},0)\cup(\sqrt{3},\infty)) and decreasing on ((-\infty,-\sqrt{3})\cup(0,\sqrt{3}))

Answer:

For local minimum: A. The local minimum/a is/are at (x=-\sqrt{3},\sqrt{3}) For inflection points: The inflection points are at (x=-1,1) (assuming there was a typo in the provided inflection - point option). For increasing and decreasing intervals: A. (f) is increasing on ((-\sqrt{3},0)\cup(\sqrt{3},\infty)) and (f) is decreasing on ((-\infty,-\sqrt{3})\cup(0,\sqrt{3}))