Question

Consider the following. f(x) = - *** + 3x3- 1/3 - 3x2 + 8 Find f'(x). f'(x) = Find F"(x). f"(x) = Find the x-values of the possible points of inflection. (Enter your answers as a comma-separated list.) X = Determine the intervals on which the function is concave up. (Enter your answer using interval notation.) Determine the intervals on which the function is concave down. (Enter your answer using interval notation.) Locate the points of inflection. (If an answer does not exist, enter DNE.) (x, y) = (smaller x-value) (x, y) = (larger x-value)

Answer 1

2 f'(x) I f R 3 x² +%* =0 2 31-724-X-250 (DE (-2,1)) f(x) = 14x4 ;*" + 1 a 3 – 3 x² + 8 a lice ( 4x4 31248) I f(x) t%2x2-62 | F"(x) 3x2 + 3x.-6 Point of Inflection, *f *"(X) = 0 7 3*?+34-6=0 x ²4 x 2 (x + 2)(x-1)=0 X=-2,1 if f(x) > 0 (x+2) (x-1)70 XE (-01-2) U(1,0) Concare down? 6 0 (x+2)(at)KO f(-2) = Yu (-234 +5 (-273-3(-2)2+8= 4-4-12+8=74 f) et Location of points of inflections Txiy) (-2,-4) (My)= (, 2) Concave up:- a) Yu + Y₂ -3 to = 34 +5 = 23