find the intervals on which the graph of ( f ) is concave upward, the intervals on which the graph of ( f )…

find the intervals on which the graph of ( f ) is concave upward, the intervals on which the graph of ( f ) is concave downward, and the inflection points.\n( f(x)=-x^{6}+12 x^{5}-12 x + 3 )\nfor what interval(s) of ( x ) is the graph of ( f ) concave upward? select the correct choice below and, if necessary, fill in the answer box to complete your choice\na. ( (0,8) )\n(type your answer in interval notation. type an exact answer. use a comma to separate answers as needed.)\nb. the graph is never concave upward.\nfor what interval(s) of ( x ) is the graph of ( f ) concave downward? select the correct choice below and, if necessary, fill in the answer box to complete your choice\na.\n(type your answer in interval notation. type an exact answer. use a comma to separate answers as needed.)\nb. the graph is never concave downward.

find the intervals on which the graph of ( f ) is concave upward, the intervals on which the graph of ( f ) is concave downward, and the inflection points.\n( f(x)=-x^{6}+12 x^{5}-12 x + 3 )\nfor what interval(s) of ( x ) is the graph of ( f ) concave upward? select the correct choice below and, if necessary, fill in the answer box to complete your choice\na. ( (0,8) )\n(type your answer in interval notation. type an exact answer. use a comma to separate answers as needed.)\nb. the graph is never concave upward.\nfor what interval(s) of ( x ) is the graph of ( f ) concave downward? select the correct choice below and, if necessary, fill in the answer box to complete your choice\na.\n(type your answer in interval notation. type an exact answer. use a comma to separate answers as needed.)\nb. the graph is never concave downward.

Answer

Explanation:

Step1: Find the first derivative

Using the power rule ((x^n)^\prime=nx^{n - 1}), for (f(x)=-x^{6}+12x^{5}-12x + 3), we have (f^\prime(x)=-6x^{5}+60x^{4}-12).

Step2: Find the second derivative

Differentiate (f^\prime(x)) again. (f^{\prime\prime}(x)=-30x^{4}+240x^{3}=-30x^{3}(x - 8)).

Step3: Find where (f^{\prime\prime}(x)>0) (concave upward)

Set (f^{\prime\prime}(x)>0), (-30x^{3}(x - 8)>0). The critical points are (x = 0) and (x = 8). Test intervals:

  • For (x<0), let (x=-1), (f^{\prime\prime}(-1)=-30(-1)^{3}(-1 - 8)=-270<0).
  • For (0<x<8), let (x = 1), (f^{\prime\prime}(1)=-30(1)^{3}(1 - 8)=210>0).
  • For (x>8), let (x = 9), (f^{\prime\prime}(9)=-30(9)^{3}(9 - 8)=-21870<0).

Step4: Find where (f^{\prime\prime}(x)<0) (concave downward)

From the above tests, the intervals where (f^{\prime\prime}(x)<0) are ((-\infty,0)\cup(8,\infty))

Answer:

For concave - upward: ((0,8)) For concave - downward: ((-\infty,0)\cup(8,\infty))