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 local minimuma is/are at x = 3/4. (type an integer or simplified fraction. use a comma to separate answers as needed.) b. there is no local minimum. what are the inflection points? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the inflection points are at x = 3/2,3. (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 and decreasing on b. f is never decreasing; f is increasing on c. f is never increasing; f is decreasing 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 the local minimuma is/are at x = 3/4. (type an integer or simplified fraction. use a comma to separate answers as needed.) b. there is no local minimum. what are the inflection points? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the inflection points are at x = 3/2,3. (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 and decreasing on b. f is never decreasing; f is increasing on c. f is never increasing; f is decreasing on

Answer

Explanation:

Step1: Find the first - derivative

First, expand (f(x)=2x(x - 3)^{3}=2x(x^{3}-9x^{2}+27x - 27)=2x^{4}-18x^{3}+54x^{2}-54x). Then, (f^\prime(x)=8x^{3}-54x^{2}+108x - 54). Factor out 2: (f^\prime(x)=2(4x^{3}-27x^{2}+54x - 27)). By the rational - root theorem and testing values, we find that (x = \frac{3}{4}) is a root. Dividing (4x^{3}-27x^{2}+54x - 27) by ((4x - 3)) gives (x^{2}-6x + 9=(x - 3)^{2}). So, (f^\prime(x)=2(4x - 3)(x - 3)^{2}). Set (f^\prime(x)=0), we get (x=\frac{3}{4},3). Test intervals: For (x\lt\frac{3}{4}), let (x = 0), then (f^\prime(0)=2(-3)(9)\lt0), so (f(x)) is decreasing on ((-\infty,\frac{3}{4})). For (\frac{3}{4}\lt x\lt3), let (x = 1), then (f^\prime(1)=2(1)(4)\gt0), so (f(x)) is increasing on ((\frac{3}{4},3)). For (x\gt3), let (x = 4), then (f^\prime(4)=2(13)(1)\gt0), so (f(x)) is increasing on ((3,\infty)). So, (f(x)) is increasing on ((\frac{3}{4},3)\cup(3,\infty)) and decreasing on ((-\infty,\frac{3}{4})).

Step2: Find the second - derivative

(f^\prime(x)=8x^{3}-54x^{2}+108x - 54), then (f^{\prime\prime}(x)=24x^{2}-108x + 108). Factor out 24: (f^{\prime\prime}(x)=24(x^{2}-\frac{9}{2}x+\frac{9}{2})). Set (f^{\prime\prime}(x)=0), using the quadratic formula (x=\frac{\frac{9}{2}\pm\sqrt{\frac{81}{4}-18}}{2}=\frac{\frac{9}{2}\pm\sqrt{\frac{81 - 72}{4}}}{2}=\frac{\frac{9}{2}\pm\frac{3}{2}}{2}). We get (x=\frac{\frac{9 + 3}{2}}{2}=3) and (x=\frac{\frac{9-3}{2}}{2}=\frac{3}{2}).

Answer:

The local minimum is at (x=\frac{3}{4}). The inflection points are at (x = \frac{3}{2},3). A. (f) is increasing on ((\frac{3}{4},3)\cup(3,\infty)) and decreasing on ((-\infty,\frac{3}{4}))