for the following function, a) give the coordinates of any critical points and classify each point as a…

for the following function, a) give the coordinates of any critical points and classify each point as a relative maximum, a relative minimum, or neither; b) identify intervals where the function is increasing or decreasing; c) give the coordinates of any points of inflection; d) identify intervals where the function is concave up or concave down; and e) sketch the graph. (g(x)=x^{3}-12x^{2}+45x + 2) a) what are the coordinates of the relative extrema? select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. a. the relative minimum point(s) is/are and the relative maximum point(s) is/are (simplify your answers. use integers or fractions for any numbers in the expression. type an ordered - pair. use a comma to separate answers as needed.) b. the relative minimum point(s) is/are and there are no relative maximum point(s) (simplify your answer. use integers or fractions for any numbers in the expression. type an ordered - pair. use a comma to separate answers as needed.) c. the relative maximum point(s) is/are and there are no relative minimum point(s) (simplify your answer. use integers or fractions for any numbers in the expression. type an ordered - pair. use a comma to separate answers as needed.) d. there are no relative minimum points and there are no relative maximum points b) on what interval(s) is (g) increasing or decreasing? select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. a. the function is increasing on the function is decreasing on (simplify your answers. type your answers in interval notation. use a comma to separate answers as needed.) b. the function is decreasing on the function is never increasing (simplify your answer. type your answer in interval notation. use a comma to separate answers as needed.) c. the function is increasing on the function is never decreasing (simplify your answer. type your answer in interval notation. use a comma to separate answers as needed.)

for the following function, a) give the coordinates of any critical points and classify each point as a relative maximum, a relative minimum, or neither; b) identify intervals where the function is increasing or decreasing; c) give the coordinates of any points of inflection; d) identify intervals where the function is concave up or concave down; and e) sketch the graph. (g(x)=x^{3}-12x^{2}+45x + 2) a) what are the coordinates of the relative extrema? select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. a. the relative minimum point(s) is/are and the relative maximum point(s) is/are (simplify your answers. use integers or fractions for any numbers in the expression. type an ordered - pair. use a comma to separate answers as needed.) b. the relative minimum point(s) is/are and there are no relative maximum point(s) (simplify your answer. use integers or fractions for any numbers in the expression. type an ordered - pair. use a comma to separate answers as needed.) c. the relative maximum point(s) is/are and there are no relative minimum point(s) (simplify your answer. use integers or fractions for any numbers in the expression. type an ordered - pair. use a comma to separate answers as needed.) d. there are no relative minimum points and there are no relative maximum points b) on what interval(s) is (g) increasing or decreasing? select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. a. the function is increasing on the function is decreasing on (simplify your answers. type your answers in interval notation. use a comma to separate answers as needed.) b. the function is decreasing on the function is never increasing (simplify your answer. type your answer in interval notation. use a comma to separate answers as needed.) c. the function is increasing on the function is never decreasing (simplify your answer. type your answer in interval notation. use a comma to separate answers as needed.)

Answer

Explanation:

Step1: Find the first - derivative

Given (g(x)=x^{3}-12x^{2}+45x + 2), then (g'(x)=3x^{2}-24x + 45). Factor out 3: (g'(x)=3(x^{2}-8x + 15)=3(x - 3)(x - 5)).

Step2: Find critical points

Set (g'(x)=0), so (3(x - 3)(x - 5)=0). Then (x = 3) and (x = 5). Substitute (x = 3) into (g(x)): (g(3)=3^{3}-12\times3^{2}+45\times3 + 2=27-108 + 135+2=56). Substitute (x = 5) into (g(x)): (g(5)=5^{3}-12\times5^{2}+45\times5 + 2=125-300 + 225+2=52). The critical points are ((3,56)) and ((5,52)).

Step3: Determine relative extrema

Use the first - derivative test. Choose test points: Let (x = 2), (g'(2)=3\times(2 - 3)\times(2 - 5)=9>0). Let (x = 4), (g'(4)=3\times(4 - 3)\times(4 - 5)=-3<0). Let (x = 6), (g'(6)=3\times(6 - 3)\times(6 - 5)=9>0). Since (g'(x)) changes sign from positive to negative at (x = 3), ((3,56)) is a relative maximum. Since (g'(x)) changes sign from negative to positive at (x = 5), ((5,52)) is a relative minimum.

Step4: Find intervals of increase and decrease

(g'(x)=3(x - 3)(x - 5)). (g'(x)>0) when (x<3) or (x > 5), so the function is increasing on ((-\infty,3)\cup(5,\infty)). (g'(x)<0) when (3<x<5), so the function is decreasing on ((3,5)).

Step5: Find the second - derivative

(g'(x)=3x^{2}-24x + 45), then (g''(x)=6x-24).

Step6: Find inflection points

Set (g''(x)=0), (6x-24 = 0), so (x = 4). Substitute (x = 4) into (g(x)): (g(4)=4^{3}-12\times4^{2}+45\times4 + 2=64-192+180 + 2=54). The inflection point is ((4,54)).

Step7: Determine concavity

(g''(x)=6x - 24). (g''(x)>0) when (x>4), so the function is concave up on ((4,\infty)). (g''(x)<0) when (x<4), so the function is concave down on ((-\infty,4)).

Answer:

a) A. The relative minimum point(s) is ((5,52)) and the relative maximum point(s) is ((3,56)) b) A. The function is increasing on ((-\infty,3)\cup(5,\infty)). The function is decreasing on ((3,5))