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 the domain is (-∞,∞). (type your answer in interval notation. use a comma to separate answers as needed.) what is the y - intercept? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the y - intercept is y = 0. b. there is no y - intercept. what is/are the x - intercept(s)? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the x - intercept(s) is/are x = 0,3. (simplify your answer. use a comma to separate answers as needed.) b. there is no x - intercept. what is/are the local maximum/a? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the local maximum/a is/are at x = (type an integer or simplified fraction. use a comma to separate answers as needed.) b. there is no local maximum.

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of y = f(x). f(x)=2x(x - 3)^3 the domain is (-∞,∞). (type your answer in interval notation. use a comma to separate answers as needed.) what is the y - intercept? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the y - intercept is y = 0. b. there is no y - intercept. what is/are the x - intercept(s)? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the x - intercept(s) is/are x = 0,3. (simplify your answer. use a comma to separate answers as needed.) b. there is no x - intercept. what is/are the local maximum/a? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the local maximum/a is/are at x = (type an integer or simplified fraction. use a comma to separate answers as needed.) b. there is no local maximum.

Answer

Explanation:

Step1: Find the derivative

First, expand (f(x)=2x(x - 3)^{3}=2x(x^{3}-9x^{2}+27x - 27)=2x^{4}-18x^{3}+54x^{2}-54x). Then, use the power - rule ((x^n)^\prime=nx^{n - 1}) to find (f^\prime(x)=8x^{3}-54x^{2}+108x - 54). We can factor out a 2: (f^\prime(x)=2(4x^{3}-27x^{2}+54x - 27)). By trying some simple values like (x = \frac{3}{2}), we find that (4(\frac{3}{2})^{3}-27(\frac{3}{2})^{2}+54(\frac{3}{2})-27=4\times\frac{27}{8}-27\times\frac{9}{4}+81 - 27=\frac{27}{2}-\frac{243}{4}+54=\frac{54 - 243+216}{4}=\frac{270 - 243}{4}=\frac{27}{4}\neq0). Using the rational - root theorem, the possible rational roots of (4x^{3}-27x^{2}+54x - 27) are factors of (\frac{27}{4}). We can also use the product rule. If (u = 2x) and (v=(x - 3)^{3}), then (u^\prime=2) and (v^\prime = 3(x - 3)^{2}). By the product rule ((uv)^\prime=u^\prime v+uv^\prime), (f^\prime(x)=2(x - 3)^{3}+6x(x - 3)^{2}=2(x - 3)^{2}[(x - 3)+3x]=2(x - 3)^{2}(4x - 3)).

Step2: Find critical points

Set (f^\prime(x)=0). Then (2(x - 3)^{2}(4x - 3)=0). Using the zero - product property, (x = 3) or (x=\frac{3}{4}).

Step3: Determine local maxima/minima

We use the first - derivative test. Consider the intervals ((-\infty,\frac{3}{4})), ((\frac{3}{4},3)) and ((3,\infty)). For (x\in(-\infty,\frac{3}{4})), let (x = 0), then (f^\prime(0)=2(0 - 3)^{2}(4\times0 - 3)=2\times9\times(-3)<0). For (x\in(\frac{3}{4},3)), let (x = 1), then (f^\prime(1)=2(1 - 3)^{2}(4\times1 - 3)=2\times4\times1>0). For (x\in(3,\infty)), let (x = 4), then (f^\prime(4)=2(4 - 3)^{2}(4\times4 - 3)=2\times1\times13>0). Since the function changes from decreasing ((f^\prime(x)<0)) to increasing ((f^\prime(x)>0)) at (x=\frac{3}{4}), (f(x)) has a local minimum at (x = \frac{3}{4}). There is no local maximum.

Answer:

B. There is no local maximum.