Use the second derivative to determine the point of inflection for y = x3 – x2 – 8x + 12. ????

y' = 3x^2 - 2x - 8

y '' = 6x - 2

at the point of inflection y'' = 0
6x-2 = 0
x = 2/6 = 1/3

if x=1/3
y = 1/27 - 1/9 - 8/3 + 12
= 250/27

the point of inflection is (1/3 , 250/27)

To determine the point of inflection for the function y = x^3 - x^2 - 8x + 12 using the second derivative, follow these steps:

Step 1: Find the first derivative of the function.
Take the derivative of y with respect to x to find the first derivative.

y = x^3 - x^2 - 8x + 12

dy/dx = 3x^2 - 2x - 8

Step 2: Find the second derivative of the function.
Take the derivative of the first derivative with respect to x to find the second derivative.

d^2y/dx^2 = d/dx(3x^2 - 2x - 8)
= 6x - 2

Step 3: Set the second derivative equal to zero and solve for x.
To find the potential point(s) of inflection, set the second derivative equal to zero and solve for x.

6x - 2 = 0
6x = 2
x = 2/6
x = 1/3

Step 4: Determine the y-coordinate of the point(s) of inflection.
To find the corresponding y-coordinate(s) for the point(s) of inflection, substitute the x-value into the original equation.

y = (1/3)^3 - (1/3)^2 - 8(1/3) + 12
= 1/27 - 1/9 - 8/3 + 12
= 4/27 - 8/3 + 12
= 4/27 - 72/27 + 324/27
= 256/27

So, the point of inflection for the function y = x^3 - x^2 - 8x + 12 is (1/3, 256/27).