Please find the answer

Question

f(x) = x(x − 8)³

Find the points of inflection of the graph of the function. (Order your answers from smallest to largest x, then from smallest to largest y.

Describe the concavity. (Enter your answer using interval notation

Concave Upward=

Concave Downward=

Answer 1

f(x) = x(x-8)^3

f'(x) = x(3)(x-8)^2 + (x-8)^3

f"(x) = 3x(2)(x-8) + 3(x-8)^2 + 3(x-8)^2 = 0

6x^2 -48x +6x^2 -96x +384 =0

12x^2 -144x + 384 = 0

x -12x +32 =0

(x-8)(x-4) = 0

x =4. 8 = x coordinates of the two points of inflection

y= 4(4-8)^3 and 8(8-8)^3 = 4(-4)^3 and 0

=-256 and 0

(x,y) = (4,-256), (8,0) are the inflection points

concave up in the intervals: (-infinity, 4), and (8, infinity)

concave down in (4,8)