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). f(x)=2x(x - 3)^3 (type an integer or simplified fraction. use a comma to separate answers as needed.) b. there are no inflection points. on what interval(s) is f increasing or decreasing? (type your answer in interval notation. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.) a. f is increasing on (3/4,3),(3,∞) and decreasing on (-∞,3/4) b. f is never decreasing; f is increasing on c. f is never increasing; f is decreasing on on what interval(s) is f concave upward or concave downward? (type your answer in interval notation. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.) a. f is concave upward on and concave downward on b. f is never concave downward; f is concave upward on c. f is never concave upward; f is concave downward on

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of y = f(x). f(x)=2x(x - 3)^3 (type an integer or simplified fraction. use a comma to separate answers as needed.) b. there are no inflection points. on what interval(s) is f increasing or decreasing? (type your answer in interval notation. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.) a. f is increasing on (3/4,3),(3,∞) and decreasing on (-∞,3/4) b. f is never decreasing; f is increasing on c. f is never increasing; f is decreasing on on what interval(s) is f concave upward or concave downward? (type your answer in interval notation. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.) a. f is concave upward on and concave downward on b. f is never concave downward; f is concave upward on c. f is never concave upward; f is concave downward on

Answer

Explanation:

Step1: Find the first - derivative

Use the product rule ((uv)^\prime = u^\prime v+uv^\prime), where (u = 2x) and (v=(x - 3)^3). (u^\prime=2) and (v^\prime = 3(x - 3)^2). Then (f^\prime(x)=2(x - 3)^3+6x(x - 3)^2=(x - 3)^2(2(x - 3)+6x)=(x - 3)^2(2x-6 + 6x)=(x - 3)^2(8x-6)=2(x - 3)^2(4x - 3)). Set (f^\prime(x)=0), then ((x - 3)^2(4x - 3)=0), so (x=\frac{3}{4}) or (x = 3). Test the intervals:

  • For (x<\frac{3}{4}), let (x = 0), (f^\prime(0)=2(0 - 3)^2(4\times0 - 3)=2\times9\times(-3)<0), so (f(x)) is decreasing on ((-\infty,\frac{3}{4})).
  • For (\frac{3}{4}<x<3), let (x = 1), (f^\prime(1)=2(1 - 3)^2(4\times1 - 3)=2\times4\times1>0), so (f(x)) is increasing on ((\frac{3}{4},3)).
  • For (x>3), let (x = 4), (f^\prime(4)=2(4 - 3)^2(4\times4 - 3)=2\times1\times13>0), so (f(x)) is increasing on ((3,\infty)).

Step2: Find the second - derivative

Use the product rule on (f^\prime(x)=2(x - 3)^2(4x - 3)). Let (u = 2(x - 3)^2) and (v=4x - 3). (u^\prime=4(x - 3)) and (v^\prime = 4). Then (f^{\prime\prime}(x)=4(x - 3)(4x - 3)+8(x - 3)^2=4(x - 3)(4x - 3 + 2(x - 3))=4(x - 3)(4x-3 + 2x-6)=4(x - 3)(6x - 9)=12(x - 3)(2x - 3)). Set (f^{\prime\prime}(x)=0), then (12(x - 3)(2x - 3)=0), so (x=\frac{3}{2}) or (x = 3). Test the intervals:

  • For (x<\frac{3}{2}), let (x = 1), (f^{\prime\prime}(1)=12(1 - 3)(2\times1 - 3)=12\times(-2)\times(-1)>0), so (f(x)) is concave - upward on ((-\infty,\frac{3}{2})).
  • For (\frac{3}{2}<x<3), let (x = 2), (f^{\prime\prime}(2)=12(2 - 3)(2\times2 - 3)=12\times(-1)\times1<0), so (f(x)) is concave - downward on ((\frac{3}{2},3)).
  • For (x>3), let (x = 4), (f^{\prime\prime}(4)=12(4 - 3)(2\times4 - 3)=12\times1\times5>0), so (f(x)) is concave - upward on ((3,\infty)).

Answer:

  1. For the increasing and decreasing intervals:
    • (f) is increasing on ((\frac{3}{4},3),(3,\infty)) and decreasing on ((-\infty,\frac{3}{4})), so the answer for the increasing - decreasing part is A.
  2. For the concavity intervals:
    • (f) is concave upward on ((-\infty,\frac{3}{2}),(3,\infty)) and concave downward on ((\frac{3}{2},3)), so the answer for the concavity part is A.